Content uploaded by Pascal Mossier
Author content
All content in this area was uploaded by Pascal Mossier on Feb 26, 2025
Content may be subject to copyright.
Tackling Compressible Turbulent Multi-Component
Flows with Dynamic hp-Adaptation
Pascal Mossier a,∗, Philipp Oestringer a, Jens Keim a, Catherine Mavriplisb,
Andrea D. Beck a, Claus-Dieter Munz a
aInstitute of Aerodynamics and Gas Dynamics, University of Stuttgart, Pfaffenwaldring
21, Stuttgart, 70569, Germany
bDepartment of Mechanical Engineering, University of Ottawa, Ottawa, K1N 6N5, Canada
Abstract
In this paper, we present an hp-adaptive hybrid Discontinuous Galerkin/Finite
Volume method for simulating compressible, turbulent multi-component flows.
Building on a previously established hp-adaptive strategy for hyperbolic gas- and
droplet-dynamics problems, this study extends the hybrid DG/FV approach to
viscous flows with multiple species and incorporates non-conforming interfaces,
enabling enhanced flexibility in grid generation. A central contribution of this
work lies in the computation of both convective and dissipative fluxes across
non-conforming element interfaces of mixed discretizations. To achieve accurate
shock localization and scale-resolving representation of turbulent structures, the
operator dynamically switches between an h-refined FV sub-cell scheme and a
p-adaptive DG method, based on an a priori modal solution analysis. The method
is implemented in the high-order open-source framework FLEXI and validated
against benchmark problems, including the supersonic Taylor-Green vortex and a
triplepoint shock interaction, demonstrating its robustness and accuracy for under-
resolved shock-turbulence interactions and compressible multi-species scenarios.
Finally, the method’s capabilities are showcased through an implicit large eddy
simulation of an under-expanded hydrogen jet mixing with air, highlighting its
potential for tackling challenging compressible multi-species flows in engineering.
Keywords: hp-Refinement, Multi-Species Navier–Stokes, Non-Conforming
Meshes, Supersonic Jet Flow, Compressible Large Eddy-Simulation
Email address: pascal.mossier@iag.uni-stuttgart.de (Pascal Mossier a,∗)
Preprint submitted to ResearchGate February 26, 2025
1. Introduction
Turbulent multi-component flows are relevant to a wide range of engineering
applications. A prominent example in the face of the climate crisis and the
depletion of fossil fuel reserves is the adaptation of internal combustion engines to
hydrogen. Here, a close understanding and accurate prediction of the high-speed
injection and mixing of the hydrogen with air plays a critical role in controlling
the combustion process. Simulating such flows presents significant challenges due
to the nonlinear interactions of shock waves, turbulence, acoustics, and material
interfaces, which span a wide range of spatial and temporal scales.
High-order methods are widely recognized as adept and efficient tools for
scale-resolving computations of turbulent flows, due to their exponential error
convergence for smooth problems. Prominent examples are Finite Volume (FV)
methods with weighted essentially non-oscillatory (WENO) [28, 45] and central
weighted essentially non-oscillatory (CWENO) [10] reconstructions, spectral dif-
ference methods [29] or flux reconstruction methods [19]. A particularly efficient
approach is the Discontinuous Galerkin Spectral Element Method (DGSEM) [23],
which leverages its tensor basis structure and high data locality, to allow for com-
paratively easy implementation and exceptional scalability on massively parallel
architectures.
However, compressible multi-species flows encounter a long-standing weak-
ness of DG methods, namely their tendency to produce spurious Gibbs-oscillations
in the presence of non-linear fluxes and discontinuous solutions. There exists a
panoply of strategies to cope with this issue in the literature, ranging from the addi-
tion of artificial viscosity [39, 40, 51], to filtering [26, 52], flux reconstruction [49]
and the inclusion of piece-wise constant ansatz functions [18].
The latter approach allows for recovery of sub-element information through
the inclusion of a Finite Volume sub-cell grid, as proposed by [40]. During
recent years, different flavors of this hybrid DG/FV discretization have emerged:
Sonntag and Munz presented an a priori switching between DG and FV sub-
cell discretization with a common number of degrees of freedom (DOFs) per
element. Dumbser and Loub`
ere developed the multi-dimensional optimal order
detection (MOOD) strategy [11], which replaces a DG candidate solution in an
a posteriori fashion with a WENO sub-cell reconstruction of decreasing order
until an admissible solution is found. Finally, Hennemann et al. [17] proposed a
convex combination of element local DG and FV solutions, allowing for a gradual
transition between both operators.
This paper builds on the work of Mossier et al. [38, 36, 35], who introduced an
2
hp-adaptive extension to the FV sub-cell implementation of Sonntag and Munz [47,
48]. The method enables a finer FV sub-cell resolution, decoupled from the DG
ansatz degree, effectively mitigating accuracy degradation from order reduction
through sub-cell h-refinement. In combination with a dynamic adaptation of the
DG ansatz degree, an hp-adaptive hybrid DG/FV operator was constructed. It
takes advantage of the exponential convergence of the p-adaptive DG operator
in smooth regions, as well as the excellent shock localization of the robust FV
discretization on an h-refined sub-cell grid. The resulting hp-adaptive scheme
was validated for gas dynamics benchmarks in [38] and implemented in a sharp-
interface framework, where it proved to be well suited for challenging droplet-
dynamics problems [38, 36, 37].
With the present work, the hybrid hp-adaptive DG/FV operator is extended
to viscous problems with multiple species to study compressible, turbulent multi-
component flows. Since mesh generation with hexahedral elements remains chal-
lenging, the paper generalizes element couplings to non-conforming interfaces,
facilitating flexible grid generation with local refinement. A key challenge in this
context is the computation of convective and dissipative fluxes at non-conforming
element interfaces of mixed discretizations. The scheme is implemented as an
extension to the open-source high-order code framework FLEXI [25].
We assess the performance of the novel method in handling under-resolved
turbulence in the presence of shocks with the supersonic Taylor-Green vortex
benchmark [7], and demonstrate its multi-species capabilities on non-conforming
grids through a triplepoint shock interaction problem. The scheme is finally
applied to an implicit large eddy simulation (LES) of an under-expanded H2-jet,
immersed in an air atmosphere and compared to studies of Hamzehloo [15] and
Vuorinen [50].
The paper is organized as follows: Section 2 revisits the governing contin-
uum equations for compressible multi-species flows. In Section 3, the p-adaptive
DGSEM discretization and FV sub-cell method are derived for hyperbolic-parabolic
conservation equations with both convective and dissipative fluxes. The section
focuses on the flux computation at non-conforming interfaces of variable dis-
cretizations and outlines the temporal discretization and the indicator scheme for
multi-component flows. Further, non-linear stability is addressed with a split-
form extension of the DGSEM and a positivity preserving limiter. The scheme
is validated with free-stream and convergence studies in Section 4 and applied to
a compressible Taylor-Green-Vortex and a shock-triplepoint interaction. Finally,
Section 5 presents a jet simulation involving H2and air mixing under compressible
conditions. The paper concludes with a summary and discussion in Section 6.
3
2. Governing Equations
In the present study, the Navier–Stokes equations are applied to model com-
pressible, turbulent, multi-component flows. We consider a computational domain
Ωbounded by Γ = 𝜕Ω, a physical coordinate vector 𝒙=(𝑥1, 𝑥2, 𝑥3)𝑇∈Ωand a
time interval (0, 𝑡]. Assuming a fluid with 𝑁𝑘species, the Navier–Stokes equations
are defined as
𝜕𝜌
𝜕𝑡 + ∇𝒙· ( 𝜌𝒗)=0,
𝜕𝜌𝒗
𝜕𝑡 + ∇𝒙·(𝜌𝒗⊗𝒗+𝑝I)=∇𝒙·(𝝉),
𝜕𝜌𝑒
𝜕𝑡 + ∇𝒙·[ (𝜌𝑒 +𝑝I)𝒗]=∇𝒙·(𝝉·𝒗−𝒒ℎ−𝒒𝑑),
𝜕𝜌𝑌𝑘
𝜕𝑡 + ∇𝒙· ( 𝜌𝑌𝑘𝒗)=∇𝒙·(−𝑱𝑘), 𝑘 =1, ..., 𝑁𝑘−1,
(1a)
(1b)
(1c)
(1d)
and can be expressed in flux notation as
𝜕𝒖
𝜕𝑡 + ∇𝒙·𝑭𝑐(𝒖)+ ∇𝒙·𝑭𝑣(𝒖)=0
⇔𝜕𝒖
𝜕𝑡 + ∇𝒙·(𝑭(𝒖,∇𝒙𝒖)) =0
(2a)
(2b)
with the conservative state vector 𝒖=(𝜌, 𝜌𝒗, 𝜌𝑒, 𝜌𝒀), the convective flux 𝑭𝑣and
the viscous flux 𝑭𝑣in terms of the density 𝜌, the velocity vector 𝒗=(𝑣1, 𝑣2, 𝑣3)𝑇,
the mass-specific total energy 𝑒, the pressure 𝑝and the vector of mass fractions
𝒀=𝑌1, .. ., 𝑌𝑁𝑘−1𝑇. It is sufficient to consider the first 𝑁𝑘−1 evolution equations
for the mass fractions 𝑌𝑘=𝜌𝑘
𝜌since the relationships
𝑁𝑘
𝑗=1
𝑌𝑗=1,
𝑁𝑘
𝑗=1
𝜌𝑗=𝜌(3)
have to hold for the sum of the mass fractions and partial densities. Assuming
a Newtonian fluid of multiple species 𝑁𝑘>1, Fourier’s hypothesis for heat
conduction and Fickian diffusion, the stress tensor 𝝉, heat flux 𝒒ℎand interspecies
4
enthalpy flux 𝒒𝑑are given as
𝝉=𝜇∇𝒙𝒗+ ∇𝒙𝒗𝑇−2
3(∇𝒙·𝒗)I,
𝒒ℎ=−𝜆∇𝒙𝑇,
𝒒𝑑=
𝑗
ℎ𝑗𝑱𝑗,
(4a)
(4b)
(4c)
with the dynamic viscosity 𝜇, the heat conductivity 𝜆and the species diffusion
flux 𝑱𝑘given as
𝑱𝑘=−𝜌𝐷𝑘∇𝒙𝑌𝑘−𝜌𝑌𝑘
𝑁𝑘−1
𝑗
𝐷𝑗∇𝒙𝑌𝑗,(5)
where 𝐷𝑘denotes the diffusion coefficient. The second term on the right side of
Equation (5) is a corrective term to recover Í𝑗𝑱𝑗=0 and thus ensure local mass
conservation [9].
The mass-specific total energy 𝑒comprises a specific internal energy 𝜖and a
specific kinetic energy 1
2𝒗·𝒗contribution and is given as
𝑒=𝜖+1
2𝒗·𝒗.(6)
Since the resulting equation system consists of 5 +(𝑁𝑘−1)equations for 5 +𝑁𝑘
unknowns, a caloric and a thermal equation of state (EOS) are required for closure.
They relate the density 𝜌, temperature 𝑇and concentration 𝒀to the specific internal
energy 𝜖and pressure 𝑝respectively:
𝜖=𝜖(𝜌, 𝑇 , 𝒀), 𝑝 =𝑝(𝜌, 𝑇 , 𝒀).(7)
The numerical studies in this work are restricted to mixtures of ideal gases where
the caloric and thermal EOS have the form
𝜖=𝜖(𝜌, 𝑇 , 𝒀):=Rmix
𝜅mix −1𝑇, 𝑝 =𝑝(𝜌, 𝑇 , 𝒀):=𝜌𝑇Rmix.(8)
For a mixture of ideal gases, Dalton’s law justifies the assumption of linear mixture
rules. Therefore, the ideal gas constant Rmix and the ratio of heat capacities 𝜅mix
can be expressed as
Rmix =𝑐𝑝,mix −𝑐𝑣,mix , 𝜅mix =𝑐𝑝 ,mix
𝑐𝑣,mix
(9)
5
with
𝑐𝑝,mix =
𝑁𝑘
𝑗=1
𝑌𝑗·𝑐𝑝, 𝑗 , 𝑐𝑣,mix =
𝑁𝑘
𝑗=1
𝑌𝑗·𝑐𝑣, 𝑗 .(10)
Linear mixture rules also apply for the dynamic viscosity 𝜇mix and the Prandtl
number Prmix
𝜇:=𝜇𝑣,mix =
𝑁𝑘
𝑗=1
𝑌𝑗·𝜇𝑗,Prmix :=
𝑁𝑘
𝑗=1
𝑌𝑗·Pr𝑗.(11)
Finally, the heat conductivity 𝜆mix can be computed as
𝜆mix =𝜇mix𝑐𝑝,mix
Prmix
.(12)
Throughout this paper, the species viscosity 𝜇𝑘, heat conductivities 𝜆𝑘and diffu-
sion coefficients 𝐷𝑘are assumed as constant, if not stated otherwise.
3. Numerical Method
The objective of this section is to establish an hp-adaptive hybrid discontin-
uous Galerkin method with FV sub-cell shock capturing for the simulation of
compressible turbulent multi-component flows. While the hybrid hp-adaptive
DG/FV operator was previously introduced for gas-dynamics in [38] and applied
for inviscid sharp-interface simulations in [36], the present paper extends this to
viscous multi-component flows on non-conforming grids. Therefore, the follow-
ing section is organized as follows. First, the p-adaptive DGSEM operator and the
h-refined FV operator are revisited and extended by parabolic terms. Then, the
coupling between the operators is addressed with an extension to non-conforming
grids, to facilitate a more flexible grid generation. Subsequently, the indicator
scheme is briefly discussed, accounting for the challenges of multi-component
flows. Finally, a dealiasing strategy and a positivity preserving filter are discussed.
They pertain to the non-linear stability in the presence of under-resolved turbulence
and complex shock-patterns.
3.1. hp-Adaptive Discretization
The compressible Navier–Stokes Equations (2b) are discretized on a com-
putational domain Ω⊂R3, which is subdivided into 𝐾∈Nnon-overlapping
hexahedral elements Ω𝐸such that Ω = Ð𝐾
𝑒=1Ω𝐸
𝑒and Ñ𝐾
𝑒=1Ω𝐸
𝑒=∅holds. While
6
the tensor basis elements Ω𝐸are discretized with a discontinuous Galerkin spectral
element method, each DG element Ω𝐸can be subdivided into a sub-grid of 𝑁FV
FV sub-cells 𝑒Ωper direction, in accordance with a suitable indicator function as
discussed in Section 3.4. On these FV sub-cell elements, a second-order finite
volume operator is applied. The basic idea is thus to advance an element via the
DG operator for sufficiently smooth solutions and fall back to an FV operator for
non-smooth solution features. In the following, a short recap of the DGSEM and
FV operators for the Navier–Stokes equations is provided for general ansatz de-
grees 𝑁and FV sub-cell resolutions 𝑁FV . For a detailed derivation of the DGSEM
and FV sub-cell operator, the reader is referred to [23, 21, 47].
3.1.1. DGSEM Operator
The DGSEM operator is derived for a reference element 𝐸=[−1,1]3that is
linked to the physical element Ω𝐸through a mapping from physical coordinates
𝒙=(𝑥1, 𝑥2, 𝑥3)𝑇to reference coordinates 𝝃=(𝜉1, 𝜉2, 𝜉3)𝑇. In reference space, the
Equation (2b) reads
𝐽geo
𝜕𝒖
𝜕𝑡 + ∇𝝃·F(𝒖,∇𝒖)=0,(13)
with the Jacobi determinant 𝐽geo of the mapping and the contravariant flux F. Since
Equation (13) involves gradients of the solution, we follow the lifting strategy of
Bassi and Rebay [1] and rewrite (13) as a system of first-order equations:
𝐽geo
𝜕𝒖
𝜕𝑡 + ∇𝝃·F(𝒖,𝒈)=0,
𝒈𝑑−1
𝐽geo
∇𝝃·U𝑑=0, 𝑑 =1,2,3
(14a)
(14b)
with the contravariant solution in the direction 𝑑denoted as U𝑑. By projecting
(14a) and (14b) onto a space of polynomial test functions 𝜓∈Pand integration
by parts, a weak form of both equations is obtained as
∫𝐸
(𝐽geo
𝜕𝒖
𝜕𝑡 )𝜓 𝑑Ω+∮𝜕𝐸
(F·𝒏𝝃)𝜓𝑑𝑆𝜉−∫𝐸
(F· ∇𝝃)𝜓𝑑Ω = 0,
∫𝐸
(𝐽geo 𝒈𝑑)𝜓𝑑Ω+∮𝜕𝐸
(U𝑑·𝒏𝝃)𝜓𝑑𝑆𝜉−∫𝐸
(U𝑑· ∇𝝃)𝜓𝑑Ω = 0.
(15a)
(15b)
7
Subsequently, a piece-wise polynomial ansatz is introduced for the solution 𝒖, the
contravariant flux F, the lifting variable 𝒈𝑑and the contravariant lifting flux U𝑑
𝒖(𝝃, 𝑡) ≈
𝑁
𝑖, 𝑗 ,𝑘 =0
ˆ
𝑸𝑖 𝑗 𝑘 (𝑡)𝜁𝑖 𝑗 𝑘 (𝝃),𝒈𝑑(𝝃, 𝑡) ≈
𝑁
𝑖, 𝑗 ,𝑘 =0
ˆ
𝒈𝑖 𝑗 𝑘 (𝑡)𝜁𝑖 𝑗 𝑘 (𝝃),
F(𝝃, 𝑡) ≈
𝑁
𝑖, 𝑗 ,𝑘 =0
ˆ
F𝑖 𝑗 𝑘 (ˆ
𝑸𝑖 𝑗 𝑘 )𝜁𝑖 𝑗 𝑘 (𝝃),U𝑑(𝝃, 𝑡) ≈
𝑁
𝑖, 𝑗 ,𝑘 =0
ˆ
U𝑑
𝑖 𝑗 𝑘 (𝑡)𝜁𝑖 𝑗 𝑘 (𝝃)
(16)
(17)
in a solution space spanned by tensor products of one-dimensional Lagrange
polynomials
𝜁𝑖 𝑗 𝑘 (𝝃)=ℓ𝑖(𝜉1)ℓ𝑗(𝜉2)ℓ𝑘(𝜉3).(18)
In line with the Galerkin approach, the Lagrange basis (18) is used for both the
ansatz functions 𝜁and the test functions 𝜓.
Since the piece-wise polynomial ansatz allows for discontinuities between
elements, the contravariant flux (F·𝒏𝝃)and the contravariant solution (U𝑑·𝒏𝝃)
are not uniquely defined at element interfaces. While a simple arithmetic mean
(U𝑑·𝒏𝝃) ≈ 𝑼∗=1
2(𝒖−+𝒖+),
(F𝑣·𝒏𝝃) ≈ 𝑭𝑣,∗=1
2(F𝑣(𝒖−,𝒈−) + F𝑣(𝒖+,𝒈+)),
(19)
(20)
can be assumed for the inter-element solution in the lifting Equation (15b) and the
viscous part of the contravariant flux in the main Equation (15a), the convective
flux is replaced by a characteristics based numerical flux
(F𝑐·𝒏𝝃) ≈ 𝑭𝑐,∗𝒖−,𝒖+,𝒏𝝃(21)
which is computed by a Riemann solver. Here, the superscripts −and +indicate
the surface solution from both sides at an element interface. The present study
employs the Roe solver [43] with the entropy fix of Harten and Hyman [16] as an
approximate Riemann solver when not stated otherwise.
To obtain a semi-discrete scheme, the integrals in Equations (15a) and (15b)
are replaced by a numerical quadrature. A defining feature of the DGSEM is
the collocation of interpolation and quadrature nodes, which establishes a tensor
product structure of the DGSEM operator and reduces the computational cost per
degree of freedom (DOF). In this work, either Legendre–Gauss (LG) nodes or
8
Legendre–Gauss–Lobatto (LGL) nodes are chosen. Owing to the tensor product
structure, a multi-dimensional DGSEM discretization is given as a succession of
one-dimensional operations. The semi-discrete form of (15a) and (15b) is finally
obtained as
𝜕ˆ
𝒖𝑖 𝑗 𝑘
𝜕𝑡 =−1
𝑱𝑖 𝑗 𝑘 𝑁
𝑚=0
ˆ
F1
𝑚 𝑗 𝑘 ˆ
𝑫𝑖𝑚 +[𝑭∗ˆ𝑠]+𝜉1
𝑗 𝑘 ˆ
ℓ𝑖(1) + [𝑭∗ˆ𝑠]−𝜉1
𝑗 𝑘 ˆ
ℓ𝑖(−1)
+
𝑁
𝑛=0
ˆ
F2
𝑖𝑛𝑘 ˆ
𝑫𝑗𝑛 +[𝑭∗ˆ𝑠]+𝜉2
𝑖𝑘 ˆ
ℓ𝑗(1) + [𝑭∗ˆ𝑠]−𝜉2
𝑖𝑘 ˆ
ℓ𝑗(−1)
+
𝑁
𝑜=0
ˆ
F3
𝑖 𝑗 𝑜 ˆ
𝑫𝑘𝑜 +[𝑭∗ˆ𝑠]+𝜉3
𝑖 𝑗 ˆ
ℓ𝑘(1) + [𝑭∗ˆ𝑠]−𝜉3
𝑖 𝑗 ˆ
ℓ𝑘(−1)(22)
and
ˆ
𝒈𝑑
𝑖 𝑗 𝑘 =−1
𝑱𝑖 𝑗 𝑘 𝑁
𝑚=0
ˆ
U1,𝑑
𝑚 𝑗 𝑘 ˆ
𝑫𝑖𝑚 +𝑼∗,𝑑 ˆ𝑠+𝜉1
𝑗 𝑘 ˆ
ℓ𝑖(1) + 𝑼∗,𝑑 ˆ𝑠−𝜉1
𝑗 𝑘 ˆ
ℓ𝑖(−1)
+
𝑁
𝑛=0
ˆ
U2,𝑑
𝑖𝑛𝑘 ˆ
𝑫𝑗𝑛 +𝑼∗,𝑑 ˆ𝑠+𝜉2
𝑖𝑘 ˆ
ℓ𝑗(1) + 𝑼∗,𝑑 ˆ𝑠−𝜉2
𝑖𝑘 ˆ
ℓ𝑗(−1)
+
𝑁
𝑜=0
ˆ
U3,𝑑
𝑖 𝑗 𝑜 ˆ
𝑫𝑘𝑜 +𝑼∗,𝑑 ˆ𝑠+𝜉3
𝑖 𝑗 ˆ
ℓ𝑘(1) + 𝑼∗,𝑑 ˆ𝑠−𝜉3
𝑖 𝑗 ˆ
ℓ𝑘(−1)(23)
respectively, with the abbreviations
ˆ
ℓ𝑖=ℓ𝑖
𝜔𝑖
,ˆ
𝑫𝑖 𝑗 =−𝜔𝑖
𝜔𝑗
𝑫𝑗𝑖 ,𝑫𝑖 𝑗 =𝜕ℓ 𝑗(𝜉)
𝜕𝜉 𝜉=𝜁𝑖
,(24)
denoting the weighted Lagrange polynomials ˆ
ℓ𝑖, an entry of the weighted dif-
ferentiation matrix ˆ
𝑫𝑖 𝑗 and the derivative of a Lagrange basis polynomial 𝑫𝑖 𝑗
respectively.
3.1.2. FV Sub-Cell Operator
The DGSEM’s piece-wise polynomial solution is highly accurate in smooth
regions but notoriously unstable for elements containing discontinuous solution
features like shocks, material interfaces or under-resolved gradients due to aliasing.
As a stabilization technique, Huerta et al. [18] and Persson et al. [40] proposed to
9
include piece-wise constant functions in the ansatz and test spaces. To alleviate the
accuracy loss of the low-order ansatz, the FV method is applied on a sub-grid in
each DG element Ω𝐸with 𝑁FV sub-cells per dimension 𝑑. In reference space, the
resulting equidistant grid consists of 𝑁𝑑
FV sub-cells 𝑒Ω
𝑖 𝑗 𝑘 with a characteristic length
of 𝑙FV =2
𝑁FV . Assuming a piece-wise constant basis, Equation (15a) reduces to
∫𝑒Ω
𝑖 𝑗 𝑘
(𝐽FV
geo
𝜕𝒖
𝜕𝑡 )𝑑Ω+∮𝜕𝑒Ω
𝑖 𝑗 𝑘
(F·𝒏𝝃)𝑑𝑆𝜉=0,(25a)
where 𝐽FV
geo denotes the Jacobian of the mapping to a reference sub-cell 𝑒Ω
𝑖 𝑗 𝑘 .
Subsequently, integral mean values ˆ
𝒖FV
𝑖 𝑗 𝑘 are introduced for the solution and flux
integrals are replaced by the midpoint rule. The resulting evolution equation for
the integral mean values reads as
ˆ
𝒖FV
𝑖 𝑗 𝑘
𝜕𝑡 =−1
𝐽FV
𝑖 𝑗 𝑘 [𝑭∗ˆ𝑠]𝑖−1
2𝑗 𝑘 +[𝑭∗ˆ𝑠]𝑖+1
2𝑗 𝑘
[𝑭∗ˆ𝑠]𝑖 𝑗−1
2𝑘+[𝑭∗ˆ𝑠]𝑖 𝑗+1
2𝑘
[𝑭∗ˆ𝑠]𝑖 𝑗 𝑘−1
2+[𝑭∗ˆ𝑠]𝑖 𝑗 𝑘+1
2.
(26)
As for the DG operator, the non-unique surface terms (F𝑐·𝒏𝝃)of the convective
flux contribution are replaced with a numerical flux (F𝑐·𝒏𝝃) ≈ F𝑐,∗(𝒖−,𝒖+,𝒏𝝃),
provided by a Riemann solver. In the context of the FV scheme, the second-order
derivatives in the viscous flux can be calculated as central differences.
The accuracy of the FV scheme can be further enhanced with a second-order
TVD reconstruction as proposed by Sonntag and Munz [47]. Throughout this
work, we employ a minmod function to limit the reconstruction.
3.2. Assembly of hp-adaptive Operator
For the construction of an hp-adaptive hybrid DG/FV scheme, we allow for
a variable element-local degree 𝑁∈ [𝑁min, 𝑁max]and an FV sub-cell resolution
𝑁FV, which can be chosen independently of the DG ansatz 𝑁. The bounds 𝑁min
and 𝑁max as well as the FV resolution are problem dependent and have to be
selected prior to a computation, to precompute all required geometrical metrics
and mappings between the operators. Based on a comparison of the explicit time
step conditions for the DG and FV operators, Dumbser et al. [11] proposed a sub-
cell resolution of up to 𝑁FV =2𝑁max +1 to maximize resolution without impeding
10
the overall timestep constraint. To assemble the dynamic hp-adaptive method,
three key aspects have yet to be addressed:
•To achieve dynamic refinement at runtime, the solution of a DG element
has to be transformed between different ansatz degrees 𝑁≠˜
𝑁with the
mappings 𝑽N⇝˜
Nand 𝑽˜
N⇝N.
•FV shock capturing necessitates the switching between a piece-wise polyno-
mial DG and the piece-wise constant FV representation with the mappings
𝑽FV⇝DG and 𝑽DG⇝FV.
•In the presence of variable ansatz degrees and both FV and DG elements,
the computation of convective, viscous and lifting fluxes needs to account
for non-conforming solution representations at element interfaces.
The mappings 𝑽N⇝˜
Nfor 𝑁 < ˜
𝑁can be achieved exactly through an interpola-
tion, while 𝑽˜
N⇝Nfor 𝑁 > ˜
𝑁requires a projection to the lower degree. Mapping a
polynomial solution to piece-wise constant sub-cell data, 𝑽DG⇝FV, is achieved via
integration over each sub-cell. The inverse mapping 𝑽FV⇝DG is computed as the
pseudo-inverse of 𝑽DG⇝FV, following Dumbser et al. [11]. A detailed derivation
of these mappings is performed in [47, 38].
In the following, the numerical flux computation is described for both convec-
tive and viscous contributions 𝐹∗=𝐹𝑐,∗+𝐹𝑣,∗at element interfaces with mixed
discretizations. Particular attention is devoted to the case of non-conforming
computational grids.
3.2.1. Coupling of Convective Terms
The convective component of the numerical flux 𝐹𝑐,∗is determined by solving
a Riemann problem. This necessitates a common representation of the solution at
an element interface. As previously stated in [38], this involves the transformation
to a common ansatz degree 𝑁=max(𝑁, ˜
𝑁)with 𝑽N⇝˜
Nin case of a DG interface
with variable ansatz degree.
At mixed DG/FV interfaces, the flux computation calls for the transformation
of the DG side solution to an FV representation 𝑽DG⇝FV to evaluate the flux on
the piece-wise constant FV data. Once the numerical flux (21) is computed, it is
transformed back into the native discretization of each element for application in
the surface integral via 𝑽˜
N⇝Nor 𝑽FV⇝DG respectively. Figures 1 and 2 illustrate
the flux computation for both types of mixed interfaces.
11
ˆ
𝒖𝑖 𝑗
𝒖−
𝑁
𝒖−
𝑁
𝑽N ˜
N
↦→ 𝒖−
˜
𝑁
𝑭𝑐,∗
˜
𝑁
𝑽˜
N N
↦→ 𝑭𝑐,∗
𝑁
0
𝜉1
𝜉2
DG element 𝑁=2
𝒖−
˜
𝑁
Riemann
𝒖+
˜
𝑁
0
𝜉1
𝜉2
DG element ˜
𝑁=4
Figure 1: Computation of the convective flux 𝑭𝑐,∗at element interfaces with mixed ansatz degree
𝑁≠˜
𝑁. The surface solution 𝒖−
𝑁of the left DG element is interpolated from 𝑁=2 to ˜
𝑁=4
and stored in 𝒖−
˜
𝑁. Subsequently, the convective numerical flux 𝐹𝑐, ∗(𝒖−
˜
𝑁,𝒖+
˜
𝑁)is computed on the
common ansatz degree ˜
𝑁=4. Finally, the flux 𝐹𝑐,∗is projected to the native degree 𝑁=2 of the
left element.
𝒖−
DG
𝒖−
DG
𝑽DG FV
↦→ 𝒖−
FV
𝑭𝑐,∗
FV
𝑽FV DG
↦→ 𝑭𝑐,∗
DG
0
𝜉1
𝜉2
DG element
𝒖−
FV
Riemann
𝒖+
FV
0
𝜉1
𝜉2
FV sub-cell element
Figure 2: Computation of the convective flux 𝑭𝑐 ,∗at mixed DG/FV element interfaces. The
surface solution 𝒖−
DG of the left element is projected from a polynomial representation to a piece-
wise constant FV representation and stored in 𝒖−
FV. Subsequently, the convective numerical flux
𝐹𝑐,∗(𝒖−
FV,𝒖+
FV)is computed with the piece-wise constant data and finally transformed back to the
native polynomial representation of the left element.
3.2.2. Coupling of Viscous Terms
In this work, the hybrid hp-adaptive DG/FV discretization is applied to the
full compressible Navier–Stokes equations for multi-component flows for the first
time. This involves discretizing viscous fluxes related to shear-stresses (4a), heat
conduction (4b) and species diffusion (5) and the solution of an additional lifting
Equation (14a) to obtain second derivatives.
12
𝒈−
FV
𝒈−
FV
𝑽FV DG
↦→ 𝒈−
DG
0
𝜉1
𝜉2
FV sub-cell element
𝒈−
DG
Lifting Flux
𝒈+
DG
0
𝜉1
𝜉2
DG element
Figure 3: Computation of the lifting flux 𝑼∗at mixed DG/FV element interfaces. The surface
solution 𝒈−
FV of the left element is projected a piece-wise constant FV representation to a polynomial
representation and stored in 𝒈−
DG. Subsequently, the lifting flux 𝑼∗(𝒈−
DG,𝒈+
DG)is computed with
the polynomial data for the DG element.
ˆ
𝒈𝑖 𝑗
𝒈−
𝑁
𝒈−
𝑁
𝑽N ˜
N
↦→ 𝒈−
˜
𝑁
𝑼∗
˜
𝑁
𝑽˜
N N
↦→ 𝑼∗
𝑁
0
𝜉1
𝜉2
DG element 𝑁=4
𝒈−
˜
𝑁
Lifting Flux
𝒈+
˜
𝑁
0
𝜉1
𝜉2
DG element ˜
𝑁=2
Figure 4: Computation of the lifting flux 𝑼∗at element interfaces with mixed ansatz degree 𝑁≠˜
𝑁.
The surface solution 𝒈−
𝑁of the left DG element is projected from 𝑁=4 to ˜
𝑁=2 and stored in
𝒈−
˜
𝑁. Subsequently, the lifting flux 𝑼∗(𝒈−
˜
𝑁,𝒈+
˜
𝑁)is computed on the common ansatz degree ˜
𝑁=2.
Finally, the flux 𝑼∗is interpolated to the native degree 𝑁=4 of the left element.
The viscous fluxes 𝐹𝑣,∗are evaluated analogously to the convective fluxes 𝐹𝑐 ,∗
on a common ansatz degree 𝑁=max(𝑁, ˜
𝑁)or on piece-wise constant FV data in
the case of mixed DG/FV interfaces. The sole difference to the convective fluxes
lies in the flux computation itself. Here, a simple arithmetic mean (20) proves
sufficient for the parabolic terms.
Treatment of the lifting fluxes 𝑼∗at mixed interfaces calls for a slightly different
procedure. With lifting solely performed for the DG discretization, 𝑼∗is evaluated
on a pure DG representation at mixed DG/FV interfaces by transforming the FV
13
𝜉3
𝜉1
𝜉2
(a) Mortar Type 1 and 2
𝜉3
𝜉1
𝜉2
(b) Mortar Type 3
Figure 5: Schematic of the supported mortar types with Types 1 and 2 differing only in their
orientation to the reference direction. The image illustrates the mapping of the big mortar side to
virtual small mortar sides, that allow a flux computation on conforming surface data.
side solution with 𝑽FV⇝DG to polynomial data as depicted in Figure 3. When
non-conforming ansatz degrees 𝑁≠˜
𝑁are present, the lifting fluxes are computed
on the lower degree 𝑁=min(𝑁, ˜
𝑁), as shown in Figure 4.
3.2.3. Non-Conforming Interfaces
Mesh generation with hexahedral elements remains challenging, especially for
complex geometries. Allowing for non-conforming computational grids alleviates
this to some extent and provides an efficient way to increase the spatial resolu-
tion locally. Treatment of non-conforming grids was addressed with the Mortar
Element Method (MEM) by Maday et al [32] and Mavriplis et al. [33, 12], and
further analyzed by Bernardi et al. [2, 3]. The mortar technique was later applied
to DGSEM by Kopriva [24] and Chalmers et al. [6] and adapted to mixed DG/FV
discretizations by Sonntag [46] and Krais et al. [25]. With this paragraph, we
generalize the mortar technique to cope with both p-refinement and arbitrary FV
sub-cell resolutions.
Three different non-conforming interfaces are supported and illustrated in
Figure 5. Types 1 and 2 consist of a big side connected to two small mortar sides,
each of which is exactly half of the big side in reference space. While Type 1
and 2 simply refine the small sides in different reference directions, Type 3 can
be interpreted as the combination of both and provides a two to one refinement in
both directions. Therefore, we can discuss the mortar technique based on Type 1
interfaces without loss of generality.
With the hp-adaptive operator, a panoply of different discretizations can be
combined at a single non-conforming interface. To illustrate the flux computation
at such an interface, Figure 6 depicts two possible scenarios involving different
ansatz degrees and FV sub-cell elements. The flux computation between non-
conforming elements of different discretizations requires in general five key steps:
14
u+
0u+
0
FV
FV
u−
1
𝑴FV
01 ·
u+
1
Riemann
𝑴FV
02 ·
u+
2
𝑽DG⇝FV ·
u−
2
DG
u−
2
Riemann
(a)
u+
0u+
0
DG
FV
u−
1
𝑴01 ·
𝑽DG⇝FV
u+
1
Riemann
𝑴02 ·
u+
2
𝑽N⇝˜
N·
u+
2
DG
u−
2
Riemann
(b)
Figure 6: Flux computation at a non-conforming element interface with mixed hp-discretizations.
Setup 6a shows a mortar interface where the larger element uses the FV sub-cell discretization,
while Setup 6b depicts a case with a DG discretization for the larger element.
1. The surface solution of the big mortar side 𝒖+
0is interpolated to virtual small
sides 𝒖+
1and 𝒖+
2with the Vandermonde matrices 𝑴01 and 𝑴02 . When the
surface solution uses a piece-wise constant FV sub-cell representation, the
mappings 𝑴FV
01 and 𝑴FV
02 are used. The matrices can be precomputed at the
start of a computation and are derived in [46, 25].
2. In the presence of variable discretizations, the solutions 𝒖+
1and 𝒖+
2of the
virtual small sides and the solution of the small sides 𝒖−
1and 𝒖−
2have to
be transformed to a common solution representation. This is done with the
mappings 𝑽DG⇝FV,𝑽FV⇝DG ,𝑽˜
N⇝Nand 𝑽N⇝˜
Naccording to the procedure
described in Sections 3.2.1 and 3.2.2.
3. With surface data available in a conforming representation, the flux 𝑭∗=
𝑭𝑣,∗+𝑭𝑐,∗can be computed.
4. Before applying the flux in the surface integral, it has to be transformed back
to its original discretization using 𝑽DG⇝FV,𝑽FV⇝DG ,𝑽˜
N⇝Nor 𝑽N⇝˜
N.
5. Finally, the flux is projected from the small virtual mortar sides to the big
mortar side with 𝑭∗
0=𝑴10 ·𝑭∗
1+𝑴20 ·𝑭∗
2or 𝑭∗
0=𝑴FV
10 ·𝑭∗
1+𝑴FV
20 ·𝑭∗
2
when FV sub-cell elements are involved.
15
3.3. Temporal Discretization
To obtain a fully discrete scheme, the DG/FV operator is integrated in time
with an explicit fourth-order low-storage Runge–Kutta (RK) scheme of Kennedy
and Carpenter [22]. The explicit RK scheme is subject to the Courant Friedrichs
Lewy (CFL) condition, which imposes time step constraints Δ𝑡𝑐and Δ𝑡𝑣related
to the maximum hyperbolic signal velocities 𝜆𝑐and parabolic eigenvalues 𝛿𝑣. For
DG elements of degree 𝑁and FV sub-cell elements they are given as
Δ𝑡𝑐
DG =CFL𝑐·𝛼RK (𝑁)Δ𝑥DG
𝜆𝑐(2𝑁+1),
Δ𝑡𝑐
FV =CFL𝑐·𝛼RK (0)Δ𝑥FV
𝜆𝑐,CFL𝑐∈ (0,1],
Δ𝑡𝑣
DG =CFL𝑣·𝛽RK (𝑁)(Δ𝑥DG)2
𝛿𝑣(2𝑁+1),
Δ𝑡𝑣
FV =CFL𝑣·𝛽RK (0)(Δ𝑥FV)2
𝛿𝑣,CFL𝑣∈ (0,1],
(27)
(28)
(29)
(30)
with the DG and FV sub-cell grid spacings Δ𝑥DG and Δ𝑥FV and the empirical
scaling factors for the stability region 𝛼RK and 𝛽RK, which depend on the local
ansatz degree. The coupled time step for the hyperbolic-parabolic system is finally
obtained in terms of Δ𝑡𝑐and Δ𝑡𝑣as
Δ𝑡=1
1
min
Ω{Δ𝑡𝑐}+1
min
Ω{Δ𝑡𝑣}
.(31)
3.4. Indicator Strategy
Control over both p-adaptation and FV sub-cell shock capturing is provided by
an error estimator that analyses the decay rate of the modal polynomial solution
representation [33, 34, 38]. For a given element 𝐸, the element-local solution
𝒖(𝝃)can be represented as an infinite series of polynomial basis functions 𝜁𝑖 𝑗 𝑘 (𝝃)
𝒖(𝝃)=
∞
𝑖, 𝑗 ,𝑘 =0
ˆ
𝒖𝑖 𝑗 𝑘 𝜁𝑖 𝑗 𝑘 (𝝃)=
𝑁
𝑖, 𝑗 ,𝑘 =0
ˆ
𝒖𝑖 𝑗 𝑘 𝜁𝑖 𝑗 𝑘 (𝝃)
| {z }
Ansatz
+
∞
𝑖, 𝑗 ,𝑘 =𝑁+1
ˆ
𝒖𝑖 𝑗 𝑘 𝜁𝑖 𝑗 𝑘 (𝝃)
| {z }
Truncation error
.(32)
It is evident from Equation (32) that an ansatz up to degree 𝑁is associated
with a truncation error. When transforming (32) to a modal Legendre basis with
16
the Vandermonde matrix 𝑽Leg, the new coefficients ˆ
𝒖Leg,𝑖 𝑗 𝑘 correspond to the
amplitudes of the respective solution modes. The idea of the indicator is to infer
an error estimate from the decay rate of these solution modes. Given a smooth
solution, an exponential decay of the solution modes can be expected, while the
extrapolation of the decay rate correlates with the truncation error. Oscillatory
solution behavior manifests as large amplitude of the highest solution modes, which
leads to a very slow decay or even an increased contribution of the higher modes.
The contribution 𝑤𝜉𝑖
𝑚of the 𝑚th solution mode in 𝜉𝑖-direction can be evaluated as
𝑤𝜉𝑖
𝑚=
Í𝑁
𝑗,𝑘 =0ˆ
𝒖2
Leg,𝑖 𝑗 𝑘 𝑖=𝑚
Í𝑁
𝑖, 𝑗 ,𝑘 =0ˆ
𝒖2
Leg,𝑖 𝑗 𝑘
.(33)
Subsequently, the decay rate 𝜎𝑖in the 𝑖th direction is obtained with a least-squares
fit of the contributions 𝑤𝜉𝑖
𝑚to an exponential function
𝑤𝜁𝑖
𝑚=𝑐𝑒−𝜎𝑖𝑚with 𝑐∈R.(34)
The final indicator value is obtained as the minimum over all spatial dimensions
I=min
Ω(|𝜎1|,|𝜎2|,|𝜎3|).(35)
The indicator function Iserves two purposes: it controls FV shock capturing
and determines the local ansatz degree in the p-adaptive DG operator. Switching
between ansatz degrees and shock capturing relies on evaluating and comparing I
against empirical thresholds T, as detailed in [38, 36]. To detect and distinguish
solution features that require either shock capturing or p-refinement, two control
mechanisms are available: first, the selection of variables on which Iis evaluated
– pressure 𝑝for shocks, density 𝜌for shear layers, and mass fraction 𝒀for material
interfaces – and the tuning of the thresholds T, which are separately defined for
shock capturing and p-refinement. Since shocks and sharp gradients in the species
concentration can trigger spurious oscillations, shock capturing relies on I(𝑝)
and I(𝒀). On the other hand, p-adaptation is controlled by I(𝜌)to track contact
discontinuities at shear layers. Hence, the indicators I
p-adapt for p-adaptation and
I
h-refine for h-refined sub-cell limiting are implemented as follows:
I
p-adapt :=I(𝜌),I
h-refine (𝑝, 𝒀):=min(I (𝑝),I(𝒀)).(36)
17
3.5. Dealiasing and Limiting
The numerical quadrature of the non-linear flux function, using collocation,
introduces integration errors, which may lead to aliasing and thus degrade stability
in the presence of under-resolved turbulence. To overcome this issue, we apply
the non-linearly stable split-flux form of Gassner et al. [14], which is derived from
the strong form of the DGSEM on LGL nodes. The split-form is constructed
such that it fulfills the summation-by-parts (SBP) property, a discrete analogue to
integration by parts. Using the SBP property and replacing the fluxes by suitable
two-point fluxes, the split-form DG allows to preserve properties like kinetic energy
at a discrete level, independently of the quadrature error. In the present paper,
kinetic energy preserving two-point fluxes of Pirozzoli [42] are employed. The
split-operator can be written in an equivalent form to the semi-discrete operators
(22), albeit with the fluxes of the Navier–Stokes equations replaced by two-point
fluxes [25]. Therefore, the coupling strategy at element interfaces is unaffected by
the choice between the standard DGSEM or split-form DGSEM.
While the split-form and the FV shock-capturing greatly enhance robustness
and shock-localization, a formal guarantee of stability is impossible in the face
of empirically tuned indicator thresholds Tand even a second-order FV recon-
struction in multiple space-dimensions. Thus, we additionally apply the positivity
preserving (PP) limiter of Zhang and Shu [52] to ensure positive density and
pressure during computations. The relative amount of elements affected by the
PP-limiter will be addressed in the numerical application section, to demonstrate
that this additional stabilization is indeed only required for a negligible number of
elements.
3.6. Dynamic Load Balancing
Dynamic hp-adaptation at runtime introduces variable computational costs
per element and thus workload imbalances among processors in parallel execu-
tion. For an efficient implementation of the adaptive discretization on massively
parallel high-performance computers, an efficient dynamic load balancing (DLB)
scheme is essential. DLB ensures an even distribution of computational workloads
throughout the computation and relies on three central building blocks:
1. An estimation of the current workload per processor unit P.
2. A partitioning algorithm that divides the domain Ωinto subdomains ΩP,
ensuring a balanced workload distribution across the processors P𝑛for
𝑛∈ [1, 𝑁procs ].
18
3. A heuristic for selecting a set of time steps 𝜎={𝑡0, 𝑡1, ..., 𝑡end }, referred to
as a scenario, at which dynamic load balancing is to be performed.
In the present work, we utilize the DLB implementation proposed in [36, 35].
The method estimates the workload per element for each discretization through
an initial calibration run at the start of the computation. During this calibration,
the spatial operator is evaluated once for a purely finite volume discretization O𝐹𝑉
and for discontinuous Galerkin discretizations O𝐷𝐺 (𝑁)with polynomial degrees
𝑁=𝑁min, ..., 𝑁max, yielding the element weights
𝑤(O𝐹𝑉 ), 𝑤(O𝐷𝐺 (𝑁min )), ..., 𝑤(O𝐷 𝐺 (𝑁max)).(37)
The workload per processor 𝐵𝑛is then obtained as the sum of the element weights
𝐵𝑛=Í𝑁elems
𝑖=1𝑤𝑖over the sub-partitions, assigned to the respective processor P𝑛.
Partitioning of the domain Ωinto subdomains ΩPis achieved by solving
a chains-on-chains partitioning problem [41] along a space filling curve. The
quality of the partitioning can be described by the imbalance I, which is defined
as
I=max
𝑛∈[1,𝑁procs ]𝐵𝑛
𝐵∗−1I ∈ (0,∞] (38)
with the target weight per partition 𝐵∗. Thereby, a value of I=0 corresponds to
an ideal partition.
Finally, a scenario 𝜎is chosen based on a simple heuristic where DLB is
performed at most every SDLB time steps, when the imbalance exceeds a threshold
of I>T
DLB. The performance of the described DLB strategy is discussed in
Section 5 with a study of the imbalances and a scaling test.
4. Numerical Validation
Within this section, key properties of the proposed numerical scheme like
free-stream preservation and the experimental order of convergence are analyzed.
Further, the method is applied to the well-know compressible Taylor-Green Vortex
benchmark to investigate its capability in handling under-resolved compressible
turbulence. Finally, we study a multi-component shock-triplepoint interaction on
a non-conforming grid.
4.1. Free-Stream Preservation
This section validates the proposed hp-adaptive discretization by demonstrating
its ability to preserve a uniform free-stream solution on a curved, non-conforming
19
FV
DG,N=2
DG,N=3
DG,N=4
DG,N=5
Resolution
Figure 7: Distorted cubic domain of the free-stream preservation test. The upper half uses a higher
resolution, leading to a non-conforming mesh. A random distribution of DG and FV elements is
used with an ansatz degree 𝑁=[2,5]and an FV sub-cell resolution of 𝑁FV =8 per direction.
grid with varying local element discretizations. An initial free-stream solution
𝒖=(1,1,1,1,1)𝑇is imposed on a computational domain Ω, which is defined as
a distorted periodic cube with the mapping:
𝒙↦→ 𝒙+0.1·sin(𝑥1) · sin(𝑥2) · sin(𝑥3),for 𝒙∈ [−1,1]3.(39)
The lower half of the cube features twice the resolution than the upper half,
resulting in a non-conforming grid. A polynomial ansatz of degree 𝑁𝑔𝑒𝑜 =2 is
chosen to approximate the physical geometry and the solution is discretized with
a variable ansatz degree 𝑁∈[2,5]and FV elements with a sub-cell resolution
of 𝑁FV =8. To ensure the most rigorous testing, we distribute the ansatz degree
and FV elements randomly in space and time. The resulting setup is visualized
in Figure 7. The simulation is conducted for 𝑡∈ [0,0.5]which amounts to 180
time steps. A subsequent analysis of the L2and L∞error norms, listed in Table 1,
indicates errors are within the order of machine precision. Thus, we can conclude
that the proposed dynamic adaptive scheme achieves free-stream preservation on
non-conforming grids.
𝑢1𝑢2𝑢3𝑢4𝑢5
L21.42𝑒−14 1.13𝑒−14 1.15𝑒−14 1.16𝑒−14 4.36𝑒−14
L∞3.91𝑒−14 2.88𝑒−14 2.96𝑒−14 3.03𝑒−14 9.81𝑒−14
Table 1: 𝐿2and 𝐿∞free-stream error with a hybrid DG/FV discretization after 180 time steps.
20
FV DG𝑁=2DG𝑁=3DG𝑁=4DG 𝑁=5
Resolution
1.8 2.0 2.2
Density
(a) 𝑁=[2,4](b) 𝑁=2, 𝑁FV =7 (c) Solution
Figure 8: Distorted cubic domain of the convergence test. Figures 8a and Figure 8b depict the
checkerboard distribution of ansatz degrees and FV elements respectively. Figure 8c shows the
density wave, which is propagated through the domain.
4.2. Experimental Order of Convergence
To establish the proposed method on non-conforming grids, we show an ex-
perimental analysis of the error convergence. The convergence study considers
a smooth density wave, which is transported diagonally along the axis [1,1,1]𝑇
within a distorted cubic domain analogue to that in Section 4.1.
A non-conforming discretization is chosen by prescribing twice the resolution
in the 𝑥1and 𝑥2directions for the lower half of the domain, 𝑥3<0. Figure 8
illustrates the discretized domain Ωand the solution at 𝑡=0.
To obtain a representative study of the hp-adaptive operator, different combina-
tions of element local discretizations are chosen: first, the p-adaptive DG operator
is applied for three checkerboard-like distributions of the ansatz degree 𝑁=[2,4],
𝑁=[3,5]and 𝑁=[4,6]. Then, a hybrid DG/FV discretization is chosen with
FV elements positioned in a checkerboard pattern and a resolution of 𝑁=2 and
𝑁FV =7. To avoid interference of the temporal discretization error, the CFL num-
ber is reduced to 0.5. The convergence study is performed for the standard DG
scheme on Legendre–Gauss nodes and repeated for the split-flux DG scheme on
Legendre–Gauss–Lobatto nodes. In Figure 9, the resulting L2error norms in the
density, 𝜌, are plotted as a function of the grid size Δ𝑥. A comparison against the
expected order of convergence 𝑝=𝑁+1 of the truncation error 𝑒∝𝑘(Δ𝑥)𝑃for
a smooth polynomial discretization of degree 𝑁indicates the desired convergence
behavior has been achieved. Consequently, the convergence order matches the
lowest present discretization in the setup for all considered combinations.
21
4.3. Compressible Taylor-Green Vortex
A well-know challenge for high-order methods is to achieve both accuracy and
robustness in the presence of under-resolved turbulence and shocks. In the context
of the present hp-adaptive scheme, a key requirement is the reliable distinction
between turbulent structures, which are to be treated by p-refinement, and shocks,
where the h-refined FV sub-cell operator is needed. With the supersonic Taylor-
Green vortex problem, a benchmark has been established [31, 7] to assess the ability
of numerical methods to deal with progressively decreasing turbulent scales in the
presence of shocks. The supersonic TGV is defined in a periodic, cubic domain
Ω = [−𝜋, 𝜋]3with the isothermal initial conditions
𝑣1(𝒙,0)=sin (𝑥1)cos (𝑥2)cos (𝑥3),
𝑣2(𝒙,0)=−cos (𝑥1)sin (𝑥2)cos (𝑥3),
𝑣3(𝒙,0)=0.0,
𝑝(𝒙,0)=𝑝0+𝜌0
16 (𝑡𝑒𝑥𝑡𝑐𝑜𝑠(2𝑥1) + cos(2𝑥2)) (2+cos(2𝑥3)),
(40a)
(40b)
(40c)
(40d)
where the temperature 𝑇0is set to the reference temperature of the Sutherland
law 𝑇ref and a ratio of heat capacities 𝛾=1.4. To close the initial conditions,
the Mach number is set to Ma0=𝜌0
𝛾 𝑝0
B1.25 and the Reynolds number chosen
10−1100
10−8
10−6
10−4
10−2
1
2
1
3
1
5
Δ𝑥
𝐿2-Error (𝜌)
Standard DG – LG nodes
10−1100
10−8
10−6
10−4
10−2
1
2
1
3
1
5
Δ𝑥
Split-Flux DG – LGL nodes
𝑁=[2,4]𝑁=[3,5]𝑁=[4,6]𝑁=2, 𝑁FV =7
Figure 9: Convergence of the hp-adaptive operator with the split-flux DG scheme on Legendre–
Gauss nodes (left) and the standard DG scheme on Legendre–Gauss–Lobatto nodes (right).
22
0 5 10 15 20
0
0.02
0.04
0.06
0.08
0.1
0.12
t
Ek
FLEXI-hp
Reference
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
·10−2
t
Es
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
·10−3
t
Ed
0 5 10 15 20
0
20
40
60
80
100
120
140
t[s]
Average NDOF/element
NDOF
0
2
4
6
8
10
FV elements [%]
FV elements
Figure 10: Results of the TGV flow at Re0=1600 and Ma0=1.25 for 𝑡=[0,20]. The
plots evaluate the temporal evolution of the kinetic energy 𝐸𝑘and the solenoidal and dilatational
components of the kinetic energy dissipation 𝐸𝑠and 𝐸𝑑. In addition, a statistic of the relative
amount of FV elements and the average number of DOFs per element 𝑁DOF is provided.
as Re0=𝜌0
𝜇0
B1600. A non-constant dynamic viscosity is applied, following the
Sutherland law, as explained in [7].
The TGV problem is advanced until a final time 𝑡=20 and the physical
domain Ωis discretized by 323elements. The hp-adaptive discretization applies
a variable ansatz degree 𝑁∈ [2,4]and an FV sub-cell resolution of 𝑁FV =9.
To distinguish between shocks and contact discontinuities, the indicator for p-
refinement is evaluated on the pressure, while FV shock-capturing is controlled
by the indicator, operating on the density. The computation employs the standard
DGSEM on LG nodes and does not require the PP-limiter from Section 3.5.
23
Results obtained with the present scheme are compared against the reference
solution from [7], which was computed with a 6𝑡ℎ -order Targeted Essentially Non-
Oscillatory (TENO) scheme on a very fine mesh of 20483DOFs. Following [31, 7],
we evaluate the solution based on three quantities: the kinetic energy 𝐸𝑘describing
the energy carried by the turbulent motion, the solenoidal part of the kinetic
energy dissipation 𝐸𝑠, which is related to the vortical motion in the flow, and the
dilatational component of the kinetic energy dissipation 𝐸𝑑, which correlates the
onset of shocks in the flow. In Figure 10, we can observe a near perfect agreement
of 𝐸𝑘and a close agreement in 𝐸𝑠. Though 𝐸𝑑deviates significantly from the
reference solution, it is in good agreement with computations of similar resolution
shown in [7]. To assess the performance of the indicator scheme, Figure 10 also
provides a statistic of the amount of FV sub-cells and the average number of DOFs
per element. While the amount of FV sub-cells never exceeds 4%, we want to
emphasize the near perfect correlation between the amount of FV cells and 𝐸𝑑.
Both peaks in 𝐸𝑑at 2.5𝑠and 7𝑠are perfectly mirrored in time trace of the amount
of FV cells. Similarly, the average number of DOFs per element increases with 𝐸𝑠
until it caps at 𝑡=12 since the maximum allowed ansatz degree prohibits further
refinement. In conclusion, we state that the hybrid adaptive scheme proves to be
well-suited to deal with both under-resolved turbulence and shocks.
4.4. Shock-Triplepoint Interaction
Finally, the multi-component implementation is validated with a two-component
shock-triplepoint interaction problem [30, 5]. It is defined as a three-state Riemann
problem in the two-dimensional domain Ω = [0,7] × [−1.5,1.5]with the initial
conditions:
𝒖(𝒙, 𝑡 =0)=
𝜌=1.0, 𝑝 =1.0, 𝛾 =1.4 if 𝒙∈Ω1=[0,1] × [−1.5,1.5]
𝜌=1.0, 𝑝 =0.1, 𝛾 =1.4 if 𝒙∈Ω2=[1,7] × [−1.5,0]
𝜌=0.125 , 𝑝 =0.1, 𝛾 =1.5 if 𝒙∈Ω3=[1,7]×[0,1.5].
Both viscosity and heat conduction are neglected, while a constant species diffusion
coefficient 𝐷𝑘=1·10−6is assumed for the species 𝑘∈ [1,2].
The initial pressure jump results in a shock that propagates with different
speeds in the subdomains Ω2and Ω3, which causes the formation of a distinctive
shear layer at the material interface between the fluid components. The problem is
simulated for 𝑡∈ [0,6]within the domain Ωand discretized by a non-conforming
grid with a base resolution of 280 ×120 elements and two refinement levels up to
an effective resolution of 1120 ×480.
24
Ω1
Ω2
Ω3
x1
x2
Figure 11: Non-conforming mesh for the shock-triplepoint interaction with two refinement levels.
In total, the discretization uses roughly 0.25 million DG elements. Inflow
conditions are imposed on the left boundary, while all remaining boundaries
apply non-reflecting outflow conditions. Figure 11 provides an illustration of the
geometric setup. The problem is computed with a p-adaptive DG operator with
𝑁=[2,4]and an FV sub-cell resolution of 𝑁FV =9. As in the TGV problem,
p-refinement relies on the indicator evaluated on the density and shock-capturing
is triggered by the indicator operating on the pressure. However, since material
interfaces are prone to produce oscillations in high-order methods, the indicator
is additionally evaluated on the mass fraction to stabilize sharp gradients in the
concentration with the FV operator. Due to the lack of viscosity, the formation
of intricate vortical structures in the shear layers requires the added non-linear
stability of the split-form DGSEM, as discussed in Section 3.5.
Figure 12 illustrates the flow field (12a), mass fraction (12b) and the element
local discretization (12c) at the time instance 𝑡=5. Both the density field and
mass fraction show the expected results, with rich vortical structures emanating
from the entropy shear layers and the roll-up of the material interface. Figure
12c demonstrates an excellent performance of the indicators with precise shock
detection and p-refinement applied at all primary and even secondary shear layers.
Moreover, the indicator evaluation on the mass fraction successfully detects areas
with strong concentration gradients and increases stability there by placing FV sub-
cells. Here, the advantage of the increased FV sub-cell resolution 𝑁FV > 𝑁max is
demonstrated: the h-refined sub-cell grid compensates for the reduced order with
an increased number of DOFs per element. Consequently, there is no extensive
numerical dissipation apparent in the shear layer at the material interface, where
FV stabilization of strong concentration gradients is required. Finally, it can be
remarked, that the mortar interfaces do not produce any visible artifacts in the
solution, though their influence is apparent in the top and bottom of Figure 12a
where contact lines are somewhat smeared out in the less refined regions.
25
(a) Density
0.1
0.3
1.1
3.7
Density
(b) Mass fraction
0
0.5
1
Mass fraction
(c) Discretization
FV
DG𝑁=2
DG𝑁=3
DG𝑁=4
Resolution
Figure 12: Multi-component simulation of a two-phase shock-triplepoint interaction problem. The
subplots provide the density field (a) and mass fraction (b) and indicate the local resolution (c) at
the time instance 𝑡=5.0.
26
5. Application - Under-Expanded H2Injection Jet
As a challenging application for the proposed adaptive scheme, a scale-
resolving simulation of a supersonic hydrogen jet is performed. In the face of the
climate crisis and the depletion of fossil fuel reserves, hydrogen has emerged as a
promising alternative for internal combustion engines. The high-speed injection
of hydrogen into an air atmosphere plays a critical role in direct hydrogen injection
engines. However, unlike conventional liquid fuels, hydrogen’s compressibility
and low density necessitate high-pressure injection to achieve sufficient mass flow
rates. This leads to a choked nozzle flow characterized by shocks and a super-
sonic, under-expanded jet featuring distinct shock diamonds. The formation and
breakup of this jet governs the mixing of hydrogen fuel with air, making a detailed
understanding of this process essential for optimizing combustion efficiency. Sim-
ulating such flows presents significant challenges due to the nonlinear interactions
of shock waves, turbulence, and acoustics, which span a wide range of spatial and
temporal scales.
5.1. Numerical Setup
To demonstrate the ability of the presented hp-adaptive scheme to handle multi-
component jet flows, we chose a benchmark similar to Hamzehloo et al. [15], which
examines an under-expanded hydrogen jet. The setup employs a nozzle pressure
ratio (NPR) of 10 and is based on an experimental study of Ruggles and Ekoto [44].
The jet is simulated within a cylindrical domain with a diameter of 30𝐷0and a
length of 41𝐷0, where 𝐷0=1.5 mm denotes the nozzle diameter. While the
study in [15] considers a high-pressure reservoir connected to the cylindrical
domain via a convergent nozzle, we simplify the geometry by imposing a Dirichlet
inlet boundary condition on a cross section of 𝐷0=1.5 mm, using the state
𝒖(𝒙=𝒙inlet, 𝑡 =0)defined as
𝒖(𝒙=𝒙Ω, 𝑡 =0)=(𝜌, 𝑣1, 𝑣2, 𝑣3, 𝑇 , 𝑌1, 𝑌2)𝑇
=(1.1579,0.0,0.0,0.0,296,1,0)𝑇.(41)
The inlet is extended by one nozzle diameter 𝐷0into the cylindrical domain.
Boundaries surrounding the protruding inlet and the back wall of the cylindrical
domain are treated as Euler slip walls. All remaining boundaries are modeled with
supersonic outflow conditions. Initially, a state
𝒖(𝒙=𝒙inlet, 𝑡 =0)=(𝜌, 𝑣1, 𝑣2, 𝑣3, 𝑇 , 𝑌1, 𝑌2)𝑇
=(0.5115,1195.15,0.0,0.0,245.6,0,1)𝑇(42)
27
species 𝒀𝜅 𝜈 hkg
m·si𝐷hm2
siPr 𝑅hJ
Kg·Ki
Air (1,0)𝑇1.4 1.73E-5 1.E-6 0.66328 287.0
H2(0,1)𝑇1.41 0.84E-5 1.E-6 0.64301 4124.0
Table 2: Material parameters for air and H2
is imposed within the domain Ω. Table 2 lists the material parameters employed
for the H2-jet simulation. As depicted in Figure 13, the computational domain Ω
is discretized with a block-structured, curved, non-conforming hexahedral mesh
with roughly 700 thousand elements. In smooth regions, the split-form DGSEM
operator is employed with ansatz degrees between 𝑁∈ [2,4], while shocks and
sharp concentration gradients are stabilized with the FV operator on a sub-cell
grid of 𝑁FV =7 sub-cells per direction. Positive density and pressure are ensured
with the addition of the positivity preserving limiter of Zhang and Shu [52]. To
avoid grid aligned shock instabilities at the curved Mach-disks, numerical fluxes
between FV elements and mixed DG/FV interfaces are computed with a local
Lax-Friedrichs solver. The Lax-Friedrichs solver was chosen over grid-aligned
shock stabilization techniques proposed by Fleischmann et al. [13] and Chen et
al. [8], since these methods require an additional tuning parameter.
The setup is computed for 𝑡∈ (0,500 𝜇s]on 𝑁procs =16384 processors of
the high performance computing (HPC) cluster HAWK, requiring a total of 911
40𝐷0
𝐷0
30𝐷0
𝐷0
(a) 2D Slice (b) 3D View
Figure 13: Computational domain for the H2jet simulation. The dimensions of the setup are
provided in (a) and a cut through the non-conforming 3D grid is visualized in (b).
28
thousand CPU hours. Load imbalances due to the variable ansatz degree and the
sub-cell operator are dynamically balanced every SDLB =50 timesteps, whenever
an imbalance threshold of T
DLB =1.0 is exceeded. For a detailed description and
scaling analysis of the dynamic load balancing scheme, we refer to [36]. In total,
4357 load balancing steps were performed during the simulation, amounting to
3.86% percent of the total compute time.
5.2. Computational Results
The evaluation of the H2-jet simulation focuses on three central aspects: the
performance of the dynamic hp-refinement, an assessment of the effects of dynamic
load balancing and a discussion of the flow field and jet-geometry in comparison
to reference data. Following [50], we consider a non-dimensionalized timescale
𝑡∗=𝐷0
2𝑣nozzle for the under-expanded jet flow in terms of the nozzle diameter 𝐷0and
the flow velocity at the nozzle exit, 𝑣nozzle. Figure 14 shows a slice of the domain
at 𝑡≈161𝑡∗and analyses the flow field with a Schlieren image, a depiction of the
H2mass fraction, the temperature 𝑇and the Mach number.
In addition, Figure 15 shows a cut through the domain at 𝑡≈805𝑡∗. The
plot evaluates the pressure 𝑝and superimposes the vorticity 𝜔above a threshold
of 𝜔 > 0.8·106. Furthermore, cross sections of the jet flow are depicted with
slices perpendicular to jet propagation. Here, the H2mass fraction is evaluated to
indicate the width of the jet.
A comparison to results published by Hamzehloo et al. [15] shows a good
qualitative agreement of the overall flow field. This is supported by an analysis of
the position and size of the first Mach-disk and the penetration of the jet at 𝑡≈161𝑡∗
in Table 3, where a close qualitative match to the data of [15] is achieved.
Further, the high-order approach employed in this study is able to capture
significantly finer flow scales, especially within the jet due to the low numerical
dissipation of the DGSEM. This is particularly evident in the rapid development
of eddies within the entropy shear layer downstream of the first Mach disk at
NPR Mach-Disk
Height [mm]
Mach-Disk
Width [mm]
Centerline Penetration
𝑍tip [mm]
Present Study 10 3.10 1.30 31.40
Hamzehloo 10 3.09 1.34 29.70
Table 3: Jet geometry comparison between the present study and that of Hamzehloo et al. [15] at
𝑡≈161𝑡∗
29
(a) Schlieren (b) 𝐻2Mass Fraction
1.7 2.85 4.0 0.0 0.5 1.0
(c) Temperature
70 115 160 200 250 296
(d) Mach Number
0.0 0.8 1.6 2.4 3.2 3.98
Figure 14: 2D slice visualizing the flow field of the under-expanded H2-jet in air at 𝑡≈161𝑡∗with
a Schlieren plot (a), the mass fraction (b), the temperature field (c) and the Mach number (d).
𝑡≈161𝑡∗. Consequently, the flow undergoes transition to a turbulent flow earlier
within the jet than in the reference solution, promoting faster mixing of H2and air.
For a quantitative analysis of the jet-penetration, we compared the centerline
penetration over time to data from Hamzehloo et al. [15] and a N2-jet of Vuorinen
et al. [50] in Figure 16. As discussed in [50] a normalized jet-tip penetration Ztip∗
30
0.09 0.5 1.0 1.5 2.0
Pressure [MPa]
0.0 0.25 0.5 0.75 1.0
H2Mass fraction
0.0 5.0 10.0 15.0 20.0
Vorticity [106·1
s]
Figure 15: 3D illustration of the jet flow at 𝑡≈805𝑡∗. The plot shows the pressure distribution
𝑝, vorticity above a threshold 𝜔 > 0.8·106and the H2mass fraction perpendicular to the main
direction of propagation.
can be obtained as
Z∗
tip =𝑍tip
4
𝜌0
𝜌∞
∝𝑡
𝑡∗,(43)
with the nozzle density 𝜌0and the density in the undisturbed plenum 𝜌∞. Since
the present setup initiates the jet flow through an inflow condition, rather than an
actual nozzle, the value for 𝜌0was extracted from the data of [15]. Z∗
tip is roughly
proportional to 𝑡/𝑡∗, leading to a near linear relation of Z∗
tip over 𝑡/𝑡∗. For
𝑡/𝑡∗≥8, the present study is in close agreement with the reference results.
While the present computations predict a higher jet-tip penetration at the initial
stages of the simulation 𝑡/𝑡∗≈2.8, our data lies in between the curves of Vuorinen
and Hamzehloo for 2.8<𝑡/𝑡∗<8. The initial overestimation of Z∗
tip could be
the consequence of modelling the inflow using boundary conditions instead of
simulating an actual nozzle. The following phase of a slighly underpredicted jet-
31
0 2 4 6 8 10 12 14
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
pt/t0
ζ
Hamzehloo H2- NPR = 8.5
Hamzehloo H2- NPR = 10
Vuorinen N2- NPR = 8.5
hp-Flexi H2- NPR = 10
Figure 16: Normalized centerline jet-tip penetration Z∗
tip over the non-dimensional time 𝑡∗. The
present study is compared against the results for H2-jets of Hamzehloo et al. [15] and an N2-jet of
Vuorinen et al. [50].
tip penetration could be explained by the faster development of turbulent structures
due to less numerical viscosity, leading to an earlier turbulent breakup.
Finally, the hp-adaptive dynamic refinement and the dynamic load balancing
are evaluated. Figure 17 provides a snapshot of the element-local discretization and
the number of elements per processor at 𝑡∗=161. The indicators are demonstrated
to perform as intended, with turbulent mixing zones resolved with the highest
possible ansatz degree and FV sub-cells concentrated primarily at shocks and
sharp concentration gradients. To mitigate the increased local computational
costs caused by dynamic hp-refinement, DLB significantly reduces the number of
elements per processor in the affected regions, as indicated in Figure 17b.
Statistics on the element-local discretization over time and the number of ele-
ments affected by the PP-limiter are shown in figure 18. Due to the nature of the
non-conforming computational mesh, which allocates by design the majority of
elements in the turbulent regions of the jet flow, the average number of DOFs and
the proportion of FV sub-cell elements are relatively high compared to the com-
pressible TGV test case in Section 4.3. Still, a considerable degree of compression
is maintained, as the resolution remains well below the maximum of 343 DOFs
per FV sub-cell element.
As a final statistic, Figure 18 evaluates the imbalance over time I
DLB (𝑡), where
32
(a) Discretization
DG𝑁=2DG𝑁=3DG𝑁=4FV
(b) Elements per Processor
11 50 230
Figure 17: Depiction of the hp-adaptive element-local discretization and domain-decomposition
in a slice of the H2-jet simulation at 𝑡∗=161. In (a), a snapshot of the FV sub-cell element
distribution and the local ansatz degree of the p-adaptive DG operator is shown. Figure (b)
illustrates the decomposition of the domain into sub-partitions for every processor. The color
indicates the number of elements within each partiton.
012345
·10−4
0
20
40
60
80
100
120
140
160
t[s]
Average NDOF/element
NDOF
0
5
10
15
20
25
30
Elements [%]
FV Elements [%]
PP elements [0.01%]
012345
·10−4
0
2
4
6
8
10
12
14
16
t[s]
Imbalance
Figure 18: Statistics of the adaptive operator, PP-limiting and imbalance for the H2-jet simulation.
I
DLB =0 corresponds to an identical workload among all partitions as described
in [38]. Starting from an initial value of I
DLB >13, the DLB scheme effectively
reduces the imbalance to roughly I
DLB ≈1. Although this still falls short of the
optimal load distribution, it constitutes a significant improvement in efficiency.
33
5.3. Parallel Performance Analysis
For a quantitative assessment of the parallel performance, a strong scaling test
is performed on the HPC cluster LUMI, an HPE Cray EX system, featuring a high-
speed HPE Slingshot 11 interconnect and nodes equipped with 2 ×AMD EPYC
64-core processors and a memory of up to 1024 GiB. The scaling test recomputes
the H2-jet setup for the time interval 𝑡∈ [100,101]𝜇s on up to 256 compute
nodes, corresponding to 𝑁procs =32768 processor units. The parallel efficiency
124
8163264128256
#compute nodes
104105
0
0.2
0.4
0.6
0.8
1
1.2
#DOFs/processor
Eciency
Ideal eciency
Measured eciency
Eciency w/o imbalance
(a) Parallel efficiency
1 2 4 8 16 32 64 128 256
#compute nodes
128 512 2048 8192 32768
1
10
100
#processors
Speedup
Ideal speedup
Measured speedup
Speedup w/o imbalance
(b) Parallel speedup
1 2 4 8 16 32 64 128 256
#compute nodes
128 512 2048 8192 32768
0
0,5
1
1,5
2
#processors
Imbalance
Imbalance
0
2
4
6
8
10
DLB overhead [%]
DLB overhead
(c) Imbalance Iand DLB overhead
Figure 19: Strong scaling results for the H2-jet simulation over the time interval 𝑡∈ [100,101]𝜇s
on up to 256 compute nodes of the HPC cluster LUMI. The plots illustrate (a) parallel efficiency,
(b) parallel speedup, and (c) load imbalance Ialong with the computational overhead due of the
load balancing procedure. In (a) and (b), the theoretical efficiency and speedup, neglecting the
measured imbalance, are also shown to emphasize the impact of residual imbalance on scalability.
34
and parallel speedup are evaluated in Figure 19a and Figure 19b respectively, while
the average imbalance and the DLB related overhead are analyzed in Figure 19c.
For up to 64 nodes and a load of ≈104DOFs per processor, the parallel
efficiency surpasses 70%. When scaling to 256 nodes, the per-processor load falls
to ≈3000 DOFs, corresponding to an average of 22 elements. While a significant
degradation of parallel performance is observed here, the parallel efficiency still
remaining above 50%. A comparison of the decline in parallel efficiency to the
imbalance reveals a clear correlation between both trends. Between 1 and 8 nodes,
the imbalance rises significantly, followed by a more gradual increase between 8
and 64 nodes. Beyond 64 nodes, a sharp surge in the imbalance is apparent.
This behavior results from the chosen DLB scenario 𝜎, where DLB is per-
formed at most every SDLB =50 time steps, provided that the imbalance I
exceeds the threshold T
DLB =1.0. Due to this simple heuristic, DLB only takes
effect during the scaling test for node counts above 8.
An analysis of the DLB overhead reveals that the selected heuristic is subopti-
mal. For up to 64 nodes, the computation allocates less than 3% of the wall clock
time to DLB, while the parallel efficiency has already decreased by as much as
20%. This strongly suggests that increasing the frequency of DLB steps could
yield significant benefits.
To motivate future improvements in the selection of a DLB heuristic and to
better understand the causes of suboptimal parallel scaling, we recalculated the
theoretical parallel efficiency and speedup under ideal load balance by removing
the imbalance. The theoretical results, without the influence of the imbalance,
indicate ideal or even superlinear scaling similar to strong scaling results reported
for the non-adaptive FLEXI implementation by Blind et al. [4].
It is important to emphasize that the imbalance is measured and thus an ap-
proximation. Therefore, the theoretical scaling results serve only as supporting
evidence that the hp-adaptive operator itself exhibits good scaling, and that imbal-
ance is the primary factor responsible for the degradation of parallel efficiency.
6. Conclusion
In this paper, we presented a novel hp-adaptive hybrid DG/FV discretization for
compressible, turbulent multi-species flows. The scheme combines a p-adaptive
DGSEM with an FV operator on an h-refined sub-grid to achieve high-order
convergence in smooth regions and accurate localization of shocks and material
interfaces. It was implemented as an extension to the open-source code FLEXI
and applied to the compressible Navier–Stokes equations with multiple species.
35
Building on the adaptive hybrid DG/FV discretization strategy for hyperbolic
gas- and droplet-dynamics of Mossier et al. [38, 36, 35], this work contributed key
additions. First, the adaptive operator was extended to the hyperbolic-parabolic
Navier–Stokes equations by including viscous fluxes and a lifting procedure for
second order terms. Further, the method was generalized to cover non-conforming
grids, greatly simplifying mesh generation. Since various combinations of opera-
tors and resolutions may be present between non-conforming element interfaces,
the paper focuses on transforming the surface solutions and fluxes to conforming
representations to ensure consistent coupling. In addition, we address non-linear
stability in the presence of under-resolved turbulence by combining the adaptive
scheme with the split-form DGSEM of Gassner et al. [14]. Dynamic refinement
is controlled by an error estimator, based on the decay rate of solution modes. In
the present work, we promoted the idea of evaluating the indicator on multiple
variables to target specific flow features like shocks via the pressure, shear lay-
ers via the density and material interfaces via the concentration. Since stability in
multiple space dimensions can not be rigorously guaranteed for this hybrid DG/FV
scheme, the addition of a positivity preserving limiter was suggested.
The resulting scheme was validated with canonical free-stream preservation
and experimental convergence tests. With an application to the compressible
Taylor-Green vortex benchmark, we confirmed that the scheme is capable of dis-
tinguishing between shocks and under-resolved turbulence and achieves excellent
results when compared to other high-order methods in recent studies [7]. The novel
indicator strategy, operating on multiple variables, was tested with a triplepoint
shock interaction and demonstrated the desired behavior in detecting shear layers,
material interfaces and shocks distinctly.
Finally, the scheme was applied to a large-scale implicit large eddy simulation
of a supersonic underexpanded hydrogen jet mixing with air. It demonstrated both
computational efficiency and robustness on massively parallel high-performance
systems, while accurately capturing key physical features in close agreement with
reference data from the literature.
In the future, we integrate the hp-adaptive multi-species implementation with
the sharp-interface framework for multi-phase flows with phase transition of J¨
ons
et al. [20] and Mossier et al. [37]. Further, we will utilize the novel hp-adaptive
hyperbolic-parabolic operator in sharp-interface simulations to explore compress-
ible turbulence in multi-phase systems. Finally, in response to the shift of su-
percomputers toward graphics processing units, the hp-adaptive method will be
integrated with the GPU-based implementation of FLEXI, published in [27].
36
Declarations
Funding This work was funded by the European Union and has received fund-
ing from the European High Performance Computing Joint Undertaking (JU)
and Sweden, Germany, Spain, Greece, and Denmark under grant agreement No
101093393. Funding for this work was also received by the Deutsche Forschungs-
gemeinschaft (DFG, German Research Foundation) within the framework of the
research unit FOR 2895 and FOR 2687. Moreover, this research was funded by the
DFG under Germany’s Excellence Strategy EXC 2075-390740016 and the GRK
2160/2, DROPIT. Further we want to gratefully acknowledge funded by the DFG
through SPP 2410 Hyperbolic Balance Laws in Fluid Mechanics: Complexity,
Scales, Randomness (CoScaRa). The simulations were performed on the national
supercomputer HPE Apollo Systems HAWK and the EuroHPC supercomputer
LUMI. We therefore gratefully acknowledge the support of the High Performance
Computing Center Stuttgart (HLRS) for granting access to HAWK under the grant
number hpcmphas/44084 and the EuroHPC Joint Undertaking for awarding this
project access to LUMI hosted by CSC (Finland) and the LUMI consortium through
a EuroHPC Development Access call.
Conflict of interest The corresponding author states on behalf of all authors,
that there is no conflict of interest.
Code availability The open-source code FLEXI, on which all extensions are
based, is available at www.flexi-project.org under the GNU GPL v3.0 license.
Availability of data and material All data generated or analyzed during this
study are included in this published article.
References
[1] Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element
method for the numerical solution of the compressible Navier–Stokes equa-
tions. Journal of Computational Physics 131(2), 267–279 (1997). DOI
10.1006/jcph.1996.5572
[2] Bernardi, C., Debit, N., Maday, Y.: Coupling finite element and spectral
methods: first results. Mathematics of Computation 54(189), 21–39 (1990)
37
[3] Bernardi, C., Maday, Y., Patera, A.T.: A new nonconforming approach to
domain decomposition: the mortar element method. In: H. Brezis, J.L.
Lions (eds.) Non-linear Partial Differential Equations and their Applications,
vol. 11, pp. 13–51. Pitman/Wiley, London/New York (1994)
[4] Blind, M., Gao, M., Kempf, D., Kopper, P., Kurz, M., Schwarz, A., Beck, A.:
Towards exascale CFD simulations using the discontinuous Galerkin solver
FLEXI. In: High Performance Computing in Science and Engineering ’23
(in press) (2024)
[5] Breil, J., Harribey, T., Maire, P.H., Shashkov, M.: A multi-material reale
method with mof interface reconstruction. Computers & Fluids 83, 115–
125 (2013). DOI 10.1016/j.compfluid.2012.08.015. Numerical methods for
highly compressible multi-material flow problems
[6] Chalmers, N., Agbaglah, G., Chrust, M., Mavriplis, C.: A parallel hp-
adaptive high order discontinuous Galerkin method for the incompressible
Navier-Stokes equations. Journal of Computational Physics: X 2, 100023
(2019). DOI 10.1016/j.jcpx.2019.100023
[7] Chapelier, J.B., Lusher, D.J., Van Noordt, W., Wenzel, C., Gibis, T., Mossier,
P., Beck, A., Lodato, G., Brehm, C., Ruggeri, M., Scalo, C., Sandham, N.:
Comparison of high-order numerical methodologies for the simulation of
the supersonic Taylor–Green vortex flow. Physics of Fluids 36(5), 055146
(2024). DOI 10.1063/5.0206359
[8] Chen, S., Lin, B., Li, Y., Yan, C.: HLLC+: Low- Mach shock-stable HLLC-
type Riemann solver for all-speed flows. SIAM Journal on Scientific Com-
puting 42(4), B921–B950 (2020). DOI 10.1137/18M119032X
[9] Coffee, T., Heimerl, J.: Transport algorithms for premixed, laminar steady-
state flames. Combustion and Flame 43, 273–289 (1981). DOI 0.1016/0010-
2180(81)90027-4
[10] Dumbser, M., Boscheri, W., Semplice, M., Russo, G.: Central weighted
ENO schemes for hyperbolic conservation laws on fixed and moving unstruc-
tured meshes. SIAM Journal on Scientific Computing 39(6), A2564–A2591
(2017). DOI 10.1137/17M1111036
[11] Dumbser, M., Loub`
ere, R.: A simple robust and accurate a posteriori sub-cell
finite volume limiter for the discontinuous Galerkin method on unstructured
38
meshes. Journal of Computational Physics 319, 163–199 (2016). DOI
10.1016/j.jcp.2016.05.002
[12] Feng, H., Mavriplis, C., Van der Feng, R., Biswas, R.: Parallel 3D mortar
element method for adaptive nonconforming meshes. Journal of Scientific
Computing 27(1–3), 231–243 (2005). DOI 10.1007/s10915-005-9026-z
[13] Fleischmann, N., Adami, S., Hu, X.Y., Adams, N.A.: A low dissipation
method to cure the grid-aligned shock instability. Journal of Computational
Physics 401, 109004 (2020). DOI 10.1016/j.jcp.2019.109004
[14] Gassner, G.J., Winters, A.R., Kopriva, D.A.: Split form nodal discontinuous
Galerkin schemes with summation-by-parts property for the compressible
Euler equations. Journal of Computational Physics 327, 39–66 (2016). DOI
10.1016/j.jcp.2016.09.013
[15] Hamzehloo, A., Aleiferis, P.: Large eddy simulation of highly turbulent
under-expanded hydrogen and methane jets for gaseous-fuelled internal com-
bustion engines. International Journal of Hydrogen Energy 39(36), 21275–
21296 (2014). DOI 10.1016/j.ijhydene.2014.10.016
[16] Harten, A., Lax, P.D., Leer, B.v.: On upstream differencing and Godunov-
type schemes for hyperbolic conservation laws. SIAM Review 25(1), 35–61
(1983). DOI 10.1137/1025002
[17] Hennemann, S., Rueda-Ram´
ırez, A.M., Hindenlang, F.J., Gassner, G.J.: A
provably entropy stable subcell shock capturing approach for high order split
form DG for the compressible Euler equations. Journal of Computational
Physics 426, 109935 (2021). DOI 10.1016/j.jcp.2020.109935
[18] Huerta, A., Casoni, E., Peraire, J.: A simple shock-capturing technique
for high-order discontinuous Galerkin methods. International Journal for
Numerical Methods in Fluids 69, 1614 – 1632 (2012). DOI 10.1002/fld.2654
[19] Huynh, H.T.: A flux reconstruction approach to high-order schemes includ-
ing discontinuous Galerkin methods. In: 18th AIAA Computational Fluid
Dynamics Conference. American Institute of Aeronautics and Astronautics
(2007). DOI 10.2514/6.2007-4079. URL http://dx.doi.org/10.2514/6.2007-
4079
39
[20] J¨
ons, S., Munz, C.D.: Riemann solvers for phase transition in a compressible
sharp-interface method. Applied Mathematics and Computation 440, 127624
(2023). DOI 10.1016/j.amc.2022.127624
[21] Kaiser, J., Winter, J., Adami, S., Adams, N.: Investigation of interface defor-
mation dynamics during high-Weber number cylindrical droplet breakup.
International Journal of Multiphase Flow 132, 103409 (2020). DOI
10.1016/j.ijmultiphaseflow.2020.103409
[22] Kennedy, C.A., Carpenter, M.H.: Additive Runge-Kutta schemes for
convection-diffusion-reaction equations. Applied Numerical Mathematics
44(1), 139–181 (2003). DOI 10.1016/S0168-9274(02)00138-1
[23] Kopriva, D.A.: Implementing Spectral Methods for Partial Differential Equa-
tions: Algorithms for Scientists and Engineers. Springer Netherlands (2009).
DOI 10.1007/978-90-481-2261-5
[24] Kopriva, D.A., Woodruff, S.L., Hussaini, M.Y.: Computation of electro-
magnetic scattering with a non-conforming discontinuous spectral element
method. International Journal for Numerical Methods in Engineering 53(1),
105–122 (2001). DOI 10.1002/nme.394
[25] Krais, N., Beck, A., Bolemann, T., Frank, H., Flad, D., Gassner, G., Hin-
denlang, F., Hoffmann, M., Kuhn, T., Sonntag, M., Munz, C.D.: Flexi: A
high order discontinuous Galerkin framework for hyperbolic-parabolic con-
servation laws. Computers & Mathematics with Applications 81, 186–219
(2020). DOI 10.1016/j.camwa.2020.05.004
[26] Krivodonova, L.: Limiters for high-order discontinuous Galerkin meth-
ods. Journal of Computational Physics 226(1), 879–896 (2007). DOI
10.1016/j.jcp.2007.05.011
[27] Kurz, M., Kempf, D., Blind, M.P., Kopper, P., Offenh¨
auser, P., Schwarz,
A., Starr, S., Keim, J., Beck, A.: Galæxi: Solving complex compressible
flows with high-order discontinuous Galerkin methods on accelerator-based
systems. Computer Physics Communications 306, 109388 (2025). DOI
10.1016/j.cpc.2024.109388
[28] Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory
schemes. Journal of Computational Physics 115(1), 200–212 (1994). DOI
10.1006/jcph.1994.1187
40
[29] Liu, Y., Vinokur, M., Wang, Z.: Spectral difference method for unstructured
grids I: Basic formulation. Journal of Computational Physics 216(2), 780–
801 (2006). DOI 10.1016/j.jcp.2006.01.024
[30] Loub`
ere, R., Maire, P.H., Shashkov, M., Breil, J., Galera, S.: ReALE:
A reconnection-based Arbitrary-Lagrangian–Eulerian method. Jour-
nal of Computational Physics 229(12), 4724–4761 (2010). DOI
10.1016/j.jcp.2010.03.011
[31] Lusher, D.J., Sandham, N.D.: Assessment of low-dissipative shock-capturing
schemes for the compressible Taylor–Green vortex. AIAA Journal 59(2),
533–545 (2021). DOI 10.2514/1.J059672
[32] Maday, Y., Mavriplis, C.A., Patera, A.T.: Nonconforming mortar element
method: application to spectral discretizations. In: T. Chan, J. Periaux,
O.B. Widlund (eds.) Domain Decomposition Methods, pp. 392–418. SIAM,
Philadelphia, PA (1989)
[33] Mavriplis, C.: Nonconforming Discretizations and a Posteriori Error Esti-
mators for Adaptive Spectral Element Techniques. Phd thesis, Massachusetts
Institute of Technology. Dept. of Aeronautics and Astronautics. (1989)
[34] Mavriplis, C.: A posteriori error estimators for adaptive spectral element
techniques. In: P. Wesseling (ed.) Proceedings of the Eighth GAMM-
Conference on Numerical Methods in Fluid Mechanics, pp. 333–342.
Vieweg+Teubner Verlag, Wiesbaden (1990)
[35] Mossier, P.: An hp-adaptive Strategy for Compressible Droplet Dynamics
with Phase Transition. Phd thesis, University of Stuttgart (2024)
[36] Mossier, P., Appel, D., Munz, C.D., Beck, A.: An efficient hp-adaptive
strategy for a level-set ghost fluid method. Journal of Scientific Computing
97(50) (2023). DOI 10.1007/s10915-023-02363-7
[37] Mossier, P., J¨
ons, S., Chiocchetti, S., Beck, A.D., Munz, C.D.: Numeri-
cal simulation of phase transition with the hyperbolic Godunov-Peshkov-
Romenski model. Journal of Computational Physics 520, 113514 (2025).
DOI 10.1016/j.jcp.2024.113514. URL 10.1016/j.jcp.2024.113514
41
[38] Mossier, P., Munz, C.D., Beck, A.: A p-adaptive discontinuous Galerkin
method with hp-shock capturing. Journal of Scientific Computing 91(4)
(2022). DOI 10.1007/s10915-022-01770-6
[39] Neumann, J.V., Richtmyer, R.D.: A method for the numerical calculation of
hydrodynamic shocks. Journal of Applied Physics 21, 232–237 (1950)
[40] Persson, P.O.: Shock capturing for high-order discontinuous Galerkin sim-
ulation of transient flow problems. In: 21st AIAA Computational Fluid
Dynamics Conference. American Institute of Aeronautics and Astronautics
(2013). DOI 10.2514/6.2013-3061
[41] Pınar, A., Aykanat, C.: Fast optimal load balancing algorithms for 1D par-
titioning. Journal of Parallel and Distributed Computing 64(8), 974–996
(2004)
[42] Pirozzoli, S.: Numerical methods for high-speed flows. Annual Re-
view of Fluid Mechanics 43(Volume 43, 2011), 163–194 (2011). DOI
10.1146/annurev-fluid-122109-160718
[43] Roe, P.: Approximate Riemann solvers, parameter vectors, and difference
schemes. Journal of Computational Physics 43(2), 357–372 (1981). DOI
10.1016/0021-9991(81)90128-5
[44] Ruggles, A., Ekoto, I.: Ignitability and mixing of underexpanded hydrogen
jets. International Journal of Hydrogen Energy 37(22), 17549–17560 (2012).
DOI 10.1016/j.ijhydene.2012.03.063
[45] Shu, C.W.: Essentially non-oscillatory and weighted essentially non-
oscillatory schemes for hyperbolic conservation laws, pp. 325–432. Springer
Berlin Heidelberg, Berlin, Heidelberg (1998). DOI 10.1007/BFb0096355
[46] Sonntag, M.: Shape Derivatives and Shock Capturing for the Navier–Stokes
Equations in Discontinuous Galerkin Method. Phd thesis, University of
Stuttgart (2017)
[47] Sonntag, M., Munz, C.D.: Shock capturing for discontinuous Galerkin meth-
ods using finite volume subcells. Finite Volumes for Complex Applications
VII 78, 945–953 (2014). DOI 10.1007/978-3-319-05591-6 96
42
[48] Sonntag, M., Munz, C.D.: Efficient parallelization of a shock capturing
for discontinuous Galerkin methods using finite volume sub-cells. Journal
of Scientific Computing 70, 1262–1289 (2017). DOI 10.1007/s10915-016-
0287-5
[49] Vilar, F.: A posteriori correction of high-order discontinuous Galerkin
scheme through subcell finite volume formulation and flux reconstruc-
tion. Journal of Computational Physics 387, 245–279 (2019). DOI
10.1016/j.jcp.2018.10.050. URL 10.1016/j.jcp.2018.10.050
[50] Vuorinen, V., Duwig, C., Yu, J., Boersma, B.J., Larmi, M., Tirunagari, S.,
Kaario, O.: Large-eddy simulation of highly underexpanded transient gas
jets 25, 22 (2013). DOI 10.1063/1.4772192
[51] Zeifang, J., Beck, A.: A data-driven high order sub-cell artificial viscosity
for the discontinuous Galerkin spectral element method. Journal of Compu-
tational Physics 441, 110475 (2021). DOI 10.1016/j.jcp.2021.110475
[52] Zhang, X., Shu, C.W.: On positivity-preserving high order discontin-
uous Galerkin schemes for compressible Euler equations on rectangular
meshes. Journal of Computational Physics 229(23), 8918–8934 (2010).
DOI 10.1016/j.jcp.2010.08.016
43
