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Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes with Summation-by-Parts Property for Hyperbolic Conservation Laws
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Abstract
This work is focused on the entropy analysis of a semi-discrete nodal discontinuous Galerkin spectral element method (DGSEM) on moving meshes for hyperbolic conservation laws. The DGSEM is constructed with a local tensor-product Lagrange-polynomial basis computed from Legendre-Gauss-Lobatto (LGL) points. Furthermore, the collocation of interpolation and quadrature nodes is used in the spatial discretization. This approach leads to discrete derivative approximations in space that are summation-by-parts (SBP) operators. On a static mesh, the SBP property and suitable two-point flux functions, which satisfy the entropy condition from Tadmor, allow to mimic results from the continuous entropy analysis on the discrete level. In this paper, Tadmor's condition is extended to the moving mesh framework. Based on the moving mesh entropy condition, entropy conservative two-point flux functions for the homogeneous shallow water equations and the compressible Euler equations are constructed. Furthermore, it will be proven that the semi-discrete moving mesh DGSEM is an entropy conservative scheme when a two-point flux function, which satisfies the moving mesh entropy condition, is applied in the split form DG framework. This proof does not require any exactness of quadrature in the spatial integrals of the variational form. Nevertheless, entropy conservation is not sufficient to tame discontinuities in the numerical solution and thus the entropy conservative moving mesh DGSEM is modified by adding numerical dissipation matrices to the entropy conservative fluxes. Then, the method becomes entropy stable such that the discrete mathematical entropy is bounded at any time by its initial and boundary data when the boundary conditions are specified appropriately. The theoretical properties of the moving mesh DGSEM will be validated by numerical experiments for the compressible Euler equations.
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This work focuses on the accuracy and stability of high-order nodal discontinuous Galerkin (DG) methods for under-resolved turbulence computations. In particular we consider the inviscid Taylor-Green vortex (TGV) flow to analyse the implicit large eddy simulation (iLES) capabilities of DG methods at very high Reynolds numbers. The governing equations are discretised in two ways in order to suppress aliasing errors introduced into the discrete variational forms due to the under-integration of non-linear terms. The first, more straightforward way relies on consistent/over-integration, where quadrature accuracy is improved by using a larger number of integration points, consistent with the degree of the non-linearities. The second strategy, originally applied in the high-order finite difference community, relies on a split (or skew-symmetric) form of the governing equations. Different split forms are available depending on how the variables in the non-linear terms are grouped. The desired split form is then built by averaging conservative and non-conservative forms of the governing equations, although conservativity of the DG scheme is fully preserved. A preliminary analysis based on Burgers' turbulence in one spatial dimension is conducted and shows the potential of split forms in keeping the energy of higher-order polynomial modes close to the expected levels. This indicates that the favourable dealiasing properties observed from split-form approaches in more classical schemes seem to hold for DG. The remainder of the study considers a comprehensive set of (under-resolved) computations of the inviscid TGV flow and compares the accuracy and robustness of consistent/over-integration and split form discretisations based on the local Lax-Friedrichs and Roe-type Riemann solvers...
High order methods based on diagonal-norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as nodal discontinuous Galerkin methods with diagonal mass matrices. In this work, we describe how to construct discretely entropy conservative schemes to a more general class of DG methods using flux differencing, quadrature-based projections, and specific DG differentiation operators. This approach also recovers existing methods for Burgers' equation involving dense norm and generalized SBP operators without boundary nodes. Numerical experiments confirm the stability and high order accuracy of the proposed methods for the one-dimensional compressible Euler equations.
In this work we prove that the original Bassi and Rebay [F. Bassi and S. Rebay, A High Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations, Journal of Computational Physics, 131:267--279, 1997] scheme (BR1) for the discretization of second order viscous terms within the discontinuous Galerkin collocation spectral element method (DGSEM) with Gauss Lobatto (GL) nodes is stable. More precisely, we prove in the first part that the BR1 scheme preserves energy stability of the skew-symmetric advection term DGSEM discretization for the linearized compressible Navier-Stokes equations (NSE). In the second part, we prove that the BR1 scheme preserves the entropy stability of the recently developed entropy stable compressible Euler DGSEM discretization of Carpenter et al. [M. Carpenter, T. Fisher, E. Nielsen, and S. Frankel, Entropy Stable Spectral Collocation Schemes for the Navier--Stokes Equations: Discontinuous Interfaces, SIAM Journal on Scientific Computing, 36: B835-B867, 2014] for the non-linear compressible NSE, provided that the auxiliary gradient equations use the entropy variables. Both parts are presented for fully three-dimensional, unstructured curvilinear hexahedral grids. Although the focus of this work is on the BR1 scheme, we show that the proof naturally includes the Local DG scheme of Cockburn and Shu.
We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may also include parabolic terms in order to model dissipative transport processes. High order piecewise polynomials are adopted to represent the discrete solution at each time level and within each spatial control volume of the computational grid, while high order of accuracy in time is achieved by the ADER approach. In our algorithm the spatial mesh configuration can be defined in two different ways: either by an isoparametric approach that generates curved control volumes, or by a piecewise linear decomposition of each spatial control volume into simplex sub-elements. Our numerical method belongs to the category of direct Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation formulation of the governing PDE system is considered and which already takes into account the new grid geometry directly during the computation of the numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a posteriori sub-cell finite volume limiter method, in which the validity of the candidate solution produced in each cell by an unlimited ADER-DG scheme is verified against a set of physical and numerical detection criteria. Those cells which do not satisfy all of the above criteria are flagged as troubled cells and are recomputed with a second order TVD finite volume scheme. The numerical convergence rates of the new ALE ADER-DG schemes are studied up to fourth order in space and time and several test problems are simulated. Finally, an application inspired by Inertial Confinement Fusion (ICF) type flows is considered by solving the Euler equations and the PDE of viscous and resistive magnetohydrodynamics (VRMHD).
We present estimates of spectral resolution power for under-resolved turbulent Euler flows obtained with high-order discontinuous Galerkin (DG) methods. The ‘1% rule’ based on linear dispersion–diffusion analysis introduced by Moura et al. [J. Comput. Phys. 298 (2015) 695–710] is here adapted for 3D energy spectra and validated through the inviscid Taylor–Green vortex problem. The 1% rule estimates the wavenumber beyond which numerical diffusion induces an artificial dissipation range on turbulent spectra. As the original rule relies on standard upwinding, different Riemann solvers are tested. Very good agreement is found for solvers which treat the different physical waves in a consistent manner. Relatively good agreement is still found for simpler solvers. The latter however displayed spurious features attributed to the inconsistent treatment of different physical waves. It is argued that, in the limit of vanishing viscosity, such features might have a significant impact on robustness and solution quality. The estimates proposed are regarded as useful guidelines for no-model DG-based simulations of free turbulence at very high Reynolds numbers.
We present a well-balanced nodal discontinuous Galerkin (DG) scheme for
compressible Euler equations with gravity. The DG scheme makes use of
discontinuous Lagrange basis functions supported at Gauss-Lobatto-Legendre
(GLL) nodes together with GLL quadrature using the same nodes. The
well-balanced property is achieved by a specific form of source term
discretization that depends on the nature of the hydrostatic solution, together
with the GLL nodes for quadrature of the source term. The scheme is able to
preserve isothermal and polytropic stationary solutions upto machine precision
on any mesh composed of quadrilateral cells and for any gravitational
potential. It is applied on several examples to demonstrate its well-balanced
property and the improved resolution of small perturbations around the
stationary solution.
Preface.- Introduction.- Adaptive Mesh Movement in 1D.- Discretization of PDEs on Time-Varying Meshes.- Basic Principles of Multidimensional Mesh Adaption.- Monitor Functions.- Variational Mesh Adaptive Methods.- Velocity-Based Adaptive Methods.- Appendix: Sobolev Spaces.- Appendix: Arithmetic Mean Geometric Mean Inequality and Jensen's Inequality.- Bibliography.
The term verification, when applied to a flow solver, describes the process of demon-strating that the code correctly solves its governing mathematical equations. A code that has been properly verified, therefore, is in likelihood free of programming errors that affect the theoretical order-of-accuracy of the numerical algorithm. As such, code verification is an early and integral step in building confidence in the predictive capabil-ities of simulation software. Over the past several decades, the complexity of computational algorithms in simula-tion codes has grown in response to demands for high-fidelity simulations in science and engineering. State-of-the-art simulation codes often involve complex exchanges of infor-mation amongst various physics modules, each of which may solve different equations using different algorithms on different grid topologies. As simulation codes become more sophisticated, thorough verification becomes increasingly challenging and time consum-ing, yet also more essential. In this work, attention is focused on hydrodynamics codes amenable to low-Mach number combustion where acoustic effects are unimportant. In this framework, a variable-density formulation of the Navier-Stokes equations is often used due to its computational efficiency relative to fully compressible formulations. In the variable-density equations, the pressure and density are formally decoupled by defining the density through an equation-of-state (EOS) expressed in terms of transported scalars. The EOS may be given by an analytical expression, or as is common for complex reactive systems, it may be precomputed and tabulated as a function of the scalars. Tabulated state-equations are heavily used in many popular combustion models. Ex-amples include laminar flamelet models (Peters 1984, 2000), conditional moment closure (CMC) methods (Klimenko & Bilger 1999), and some transported PDF methods (Pope 1985). These combustion models are used in a variety of codes, targeting applications that include the design and optimization of engines and power systems, prediction of pollutant formation in combustion devices, and modeling and prediction of fires. While validation studies of combustion codes are routinely performed, the application of sys-tematic verification studies is less common. In particular, the ramifications of tabulated state-relationships on the convergence and accuracy of combustion codes has not been widely investigated. As the EOS in typical combustion systems is multi-dimensional and highly non-linear, its implications on code performance are not straightforward. Owing to the non-linear character of the governing equations, it is common to use iterative algorithms to solve the discretized system of equations. With these methods, solutions are iteratively refined from an initial guess until all equations are approximately satisfied at a given point and time. Iterative approaches, by definition, do not exactly satisfy the discrete equations, and therefore, unavoidably involve certain residual errors. The matter of how small residuals must become for the numerical solution to be a 156 L. Shunn and F. Ham valid approximation of the governing mathematical equations is of obvious and practical relevance. In large-scale computations it may be prohibitively expensive to converge all solvers to machine precision. A popular execution mode, therefore, is for a solver to perform a fixed number of iterations and then proceed to subsequent steps in the algorithm. The ramifications of such an approach on the evolution of the solution are difficult to conjecture. A powerful technique that can unravel this complexity and ultimately help to verify these solvers is the method of manufactured solutions (MMS). Manufactured solutions are exact solutions to a set of governing equations that have been modified with forcing terms. This concept has been around since the early days of computer codes (see, for example, Burggraf 1966) and has been more recently formalized under the name MMS in a series of papers (Roache 1998a,b, 2002; Knupp & Salari 2003; Oberkampf et al. 2004; Roy 2005). The objective of this work is to use the method of manufactured solutions to explore the effects of tabulated constitutive relationships and iteration errors on the computational performance of low-Mach number combustion codes.
Centered numerical fluxes can be constructed for compressible Euler equations
which preserve kinetic energy in the semi-discrete finite volume scheme. The
essential feature is that the momentum flux should be of the form f^m_\jph =
\tp_\jph + \avg{u}_\jph f^\rho_\jph where \avg{u}_\jph = (u_j + u_{j+1})/2
and \tp_\jph, f^\rho_\jph are {\em any} consistent approximations to the
pressure and the mass flux. This scheme thus leaves most terms in the numerical
flux unspecified and various authors have used simple averaging. Here we
enforce approximate or exact entropy consistency which leads to a unique choice
of all the terms in the numerical fluxes. As a consequence novel entropy
conservative flux that also preserves kinetic energy for the semi-discrete
finite volume scheme has been proposed. These fluxes are centered and some
dissipation has to be added if shocks are present or if the mesh is coarse. We
construct scalar artificial dissipation terms which are kinetic energy stable
and satisfy approximate/exact entropy condition. Secondly, we use entropy-
variable based matrix dissipation flux which leads to kinetic energy and
entropy stable schemes. These schemes are shown to be free of entropy violating
solutions unlike the original Roe scheme. For hypersonic flows a blended scheme
is proposed which gives carbuncle free solutions for blunt body flows.
Numerical results for Euler and Navier-Stokes equations are presented to
demonstrate the performance of the different schemes.
We develop a one-dimensional moving-mesh method for hyperbolic systems of conservation laws. This method is based on the high-resolution finite-volume “wave-propagation method”, implemented in the CLAWPACK software package. A modified system of conservation laws is solved on a fixed, uniform computational grid, with a grid mapping function computed simultaneously in such a way that in physical space certain features are tracked by cell interfaces. The method is tested on a shock-tube problem with multiple reflections where the contact discontinuity is tracked, and also on two multifluid problems where the interface between two distinct gases is tracked. One is a standard test problem and the other also involves a moving piston whose motion is also tracked by the moving mesh.
A spectral method and a fifth-order weighted essentially non-oscillatory method were used to examine the consequences of filtering in the numerical simulation of the three-dimensional evolution of nearly-incompressible, in- viscid Taylor-Green vortex flow. It was found that numerical filtering using the high-order exponential filter and low-pass filter with sharp high mode cutoff applied in the spectral simulations significantly affects the conver- gence of the numerical solution. While the conservation property of the spectral method is highly desirable for fluid flows described by a system of hyperbolic conservation laws, spectral methods can yield erroneous results and conclusions at late evolution times when the flow eventually becomes under-resolved. In particular, it is demonstrated that the enstrophy and kinetic energy, which are two integral quantities often used to evaluate the quality of numerical schemes, can be misleading and should not be used unless one can assure that the solution is sufficiently well-resolved. In addi- tion, it was shown that for the Taylor-Green vortex (for example) it is useful to compare the predictions of at least two numerical methods with different algorithmic foundations (such as a spectral and finite-difference method) in order to corroborate the conclusions from the numerical solutions when the analytical solution is not known.
In Klingenberg, Schn\"ucke and Xia (Math. Comp. 86 (2017), 1203-1232) an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method to solve conservation laws has been developed and analyzed. In this paper, the ALE-DG method will be extended to several dimensions. The method will be designed for simplex meshes. This will ensure that the method satisfies the geometric conservation law, if the accuracy of the time integrator is not less than the value of the spatial dimension. For the semi-discrete method the L2-stability will be proven. Furthermore, an error estimate which provides the suboptimal (k+1/2) convergence with respect to the L-infinity-norm will be presented, when an arbitrary monotone flux is used and for each cell the approximating functions are given by polynomials of degree k. The two dimensional fully-discrete explicit method will be combined with the bound preserving limiter developed by Zhang, Xia and Shu in (J. Sci. Comput. 50 (2012), 29-62). This limiter does not affect the high order accuracy of a numerical method. Then, for the ALE-DG method revised by the limiter the validity of a discrete maximum principle will be proven. The numerical stability, robustness and accuracy of the method will be shown by a variety of two dimensional computational experiments on moving triangular meshes.
It has been known for a long time that high order methods can be very efficient for the numerical solution of hyperbolic balance laws if their stability is ensured. A suitable way to investigate stability of linear partial differential equations is given by the energy method, relying on integration by parts. In order to transfer these methods to the semidiscrete or discrete level, summation-by-parts (SBP) operators have emerged in finite difference methods. Recently, there has been an increased interest in generalised SBP operators applicable in finite difference, finite volume, discontinuous Galerkin, and flux reconstruction methods.
Turning to nonlinear hyperbolic balance laws, the analytical theory is not as advanced as the corresponding linear one. Due to this lack of guidance, constructing stable high order numerical methods becomes extremely difficult. Nevertheless, it still seems to be appropriate to mimic properties of solutions at the continuous level discretely, i.e. to follow the basic idea of summation-by-parts operators. Since not all such features can be mimicked exactly, additional challenges arise for generalised SBP operators. Certain conservative and entropy stable discretisations using SBP operators can be interpreted as split forms of the underlying equations. Based thereon, special boundary procedures are constructed for several linear and nonlinear balance laws with possibly varying coefficients, enabling conservative and stable discretisations using general SBP operators. Although these generalised methods can be more accurate and stable than classical ones, they are in general also computationally more expensive. Thus, it is questionable whether they are worth the effort.
After extending the theory of conservative discretisations using SBP operators and symmetric numerical fluxes, the application of these methods to nonlinear balance laws such as the shallow water equations and the Euler equations is studied. While it is not clear whether entropy stable schemes can be formulated in this way for the Euler equations and general SBP operators, it is possible to construct such schemes using classical summation-by-parts operators. Following again the idea to mimic properties of the continuous level discretely, several numerical methods are investigated and new ones are developed.
Investigating fully discrete stability, the time integration method has to be taken into account. Not relying on the computationally expensive solution of implicit equations, a new procedure to prove stability of explicit Runge-Kutta methods for linear equations is presented and applied to high order, strong stability preserving methods. Moreover, it is examined why such a proof seems to be restricted to linear equations. Finally, an underlying concept of the previous investigations is studied in detail. Since the entropy plays a crucial role in the theory of hyperbolic balance laws, it has been used as a design principle of numerical methods as described before. Extending these studies, variational principles for the entropy are investigated with respect to their applicability in numerical schemes.
We present and analyze an entropy-stable semi-discretization of the Euler equations based on high-order summation-by-parts (SBP) operators. In particular, we consider general multidimensional SBP elements, building on and generalizing previous work with tensor–product discretizations. In the absence of dissipation, we prove that the semi-discrete scheme conserves entropy; significantly, this proof of nonlinear L² stability does not rely on integral exactness. Furthermore, interior penalties can be incorporated into the discretization to ensure that the total (mathematical) entropy decreases monotonically, producing an entropy-stable scheme. SBP discretizations with curved elements remain accurate, conservative, and entropy stable provided the mapping Jacobian satisfies the discrete metric invariants; polynomial mappings at most one degree higher than the SBP operators automatically satisfy the metric invariants in two dimensions. In three-dimensions, we describe an elementwise optimization that leads to suitable Jacobians in the case of polynomial mappings. The properties of the semi-discrete scheme are verified and investigated using numerical experiments.
It is well known that semi-discrete high order discontinuous Galerkin (DG) methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws (Jiang and Shu (1994) [39]) and symmetric hyperbolic systems (Hou and Liu (2007) [36]), in any space dimension and for any triangulations. However, this property holds only for the square entropy and the integrations in the DG methods must be exact. It is significantly more difficult to design DG methods to satisfy entropy inequalities for a non-square convex entropy, and/or when the integration is approximated by a numerical quadrature. In this paper, we develop a unified framework for designing high order DG methods which will satisfy entropy inequalities for any given single convex entropy, through suitable numerical quadrature which is specific to this given entropy. Our framework applies from one-dimensional scalar cases all the way to multi-dimensional systems of conservation laws. For the one-dimensional case, our numerical quadrature is based on the methodology established in Carpenter et al. (2014) [5] and Gassner (2013) [19]. The main ingredients are summation-by-parts (SBP) operators derived from Legendre Gauss–Lobatto quadrature, the entropy conservative flux within elements, and the entropy stable flux at element interfaces. We then generalize the scheme to two-dimensional triangular meshes by constructing SBP operators on triangles based on a special quadrature rule. A local discontinuous Galerkin (LDG) type treatment is also incorporated to achieve the generalization to convection–diffusion equations. Extensive numerical experiments are performed to validate the accuracy and shock capturing efficacy of these entropy stable DG methods.
We describe a unique averaging procedure to design an entropy stable dissipation operator for the ideal magnetohydrodynamic (MHD) and compressible Euler equations. Often in the derivation of an entropy conservative numerical flux function much care is taken in the design and averaging of the entropy conservative numerical flux. We demonstrate in this work that if the discrete dissipation operator is not carefully chosen as well it can have deleterious effects on the numerical approximation. This is particularly true for very strong shocks or high Mach number flows present, for example, in astrophysical simulations. We present the underlying technique of how to construct a unique averaging technique for the discrete dissipation operator. We also demonstrate numerically the increased robustness of the approximation.
Fisher and Carpenter (High-order entropy stable finite difference schemes for non-linear conservation laws: Finite domains, Journal of Computational Physics, 252:518--557, 2013) found a remarkable equivalence of general diagonal norm high-order summation-by-parts operators to a subcell based high-order finite volume formulation. This equivalence enables the construction of provably entropy stable schemes. We show that it is possible to exploit the equivalent subcell formulation even further. With the specific choice of subcell finite volume fluxes we recover split formulations of quadratic or cubic terms. Thus, by changing the subcell finite volume flux to a specific choice, we are able to generate, in a systematic way, all common split forms of the compressible Euler advection terms, such as e.g. the Ducros splitting and the Kennedy and Gruber splitting. %This equivalence holds in general for diagonal norm summation-by-parts operators. In this work, we recast the discontinuous Galerkin spectral element method with Gauss-Lobatto nodes into the subcell finite volume form. Thus, we construct a unified high-order split form DG framework. We investigate with three dimensional numerical simulations of the inviscid Taylor Green vortex and show that the new split forms enhance the stability of computations in comparison to the standard scheme when solving turbulent vortical dominated flows.
We design a novel provably stable discontinuous Galerkin spectral element
(DGSEM) approximation to solve systems of conservation laws on moving domains.
To incorporate the motion of the domain, we use an arbitrary
Lagrangian-Eulerian formulation to map the governing equations to a fixed
reference domain. The approximation is made stable by a discretization of a
skew-symmetric formulation of the problem. We prove that the discrete
approximation is stable, conservative and, for constant coefficient problems,
maintains the free-stream preservation property. We also provide details on how
to add the new skew-symmetric ALE approximation to an existing discontinuous
Galerkin spectral element code. Lastly, we provide numerical support of the
theoretical results.
We design an arbitrary high order accurate nodal discontinuous Galerkin
spectral element approximation for the nonlinear two dimensional shallow water
equations with non-constant, possibly discontinuous, bathymetry on
unstructured, possibly curved, quadrilateral meshes. The scheme is derived from
a skew-symmetric formulation of the continuous problem. We prove that this
discretisation exactly preserves the local mass and momentum. Furthermore,
combined with a special numerical interface flux function, the method exactly
preserves the entropy, which is also the total energy for the shallow water
equations. This entropy conserving scheme is the baseline for a provably
entropy stable scheme. Finally, with a particular discretisation of the
bathymetry source term we prove that the numerical approximation is
well-balanced. The proofs and derivations use skew-symmetric reformulations of
the problem to remove aliasing errors. However, as many additional terms are
introduced, the resulting skew-symmetric scheme is not computationally
tractable. Therefore, we provide an equivalent reformulation of the
skew-symmetric scheme, which restores computational efficiency. We provide
numerical examples that verify the theoretical findings and furthermore provide
an application of the scheme for a partial break of a curved dam test problem.
In this work, we design an arbitrary high order accurate nodal discontinuous Galerkin spectral element type method for the one dimensional shallow water equations. The novel method uses a skew-symmetric formulation of the continuous problem. We prove that this discretisation exactly preserves the local mass and momentum. Furthermore, we show that combined with a special numerical interface flux function, the method exactly preserves the entropy, which is also the total energy for the shallow water equations. Finally, we prove that the surface fluxes, the skew-symmetric volume integrals, and the source term are well balanced. Numerical tests are performed to demonstrate the theoretical findings.
In the development of the next generation of numerical methods for CFD, Discontinuous Galerkin methods are promising a substantial increase in efficiency and accuracy. While the particular high order methods can be very distinct, they have in common that they must rely on a high-order approximation of curved geometries to maintain their high-order of accuracy. The generation of curved meshes is thus a topic whose importance cannot be overstated, if one truly wants to apply DG methods to problems with industrial relevance. Especially aerospace applications heavily rely on complex geometries and pose high requirements to the quality of geometry representation. In this work we present several techniques to produce high order meshes, relying on linear meshes, which can be generated by standard commercial mesh generation tools. We describe the generation of the curved surface meshes and curved volume meshes, which can be particularly difficult for curved boundary layers.
This paper shows that the discontinuous Galerkin collocation spectral element method with Gauss-Lobatto points (DGSEM-GL) satisfies the discrete summation-by-parts (SBP) property and can thus be classified as an SBP-SAT (simultaneous approximation term) scheme with a diagonal norm operator. In the same way, SBP-SAT finite difference schemes can be interpreted as discontinuous Galerkin-type methods with a corresponding weak formulation based on an inner-product formulation common in the finite element community. This relation allows the use of matrix-vector notation (common in the SBP-SAT finite difference community) to show discrete conservation for the split operator formulation of scalar nonlinear conservation laws for DGSEM-GL and diagonal norm SBP-SAT. Based on this result, a skew-symmetric energy stable discretely conservative DGSEM-GL formulation (applicable to general diagonal norm SBP-SAT schemes) for the nonlinear Burgers equation is constructed.
We derive a spectrally accurate moving mesh method for mixed material interface problems modeled by Maxwellʼs or the classical wave equation. We use a discontinuous Galerkin spectral element approximation with an arbitrary Lagrangian–Eulerian mapping and derive the exact upwind numerical fluxes to model the physics of wave reflection and transmission at jumps in material properties. Spectral accuracy is obtained by placing moving material interfaces at element boundaries and solving the appropriate Riemann problem. We present numerical examples showing the performance of the method for plane wave reflection and transmission at dielectric and acoustic interfaces.
Developing stable and robust high-order finite-difference schemes requires mathematical
formalism and appropriate methods of analysis. In this work, nonlinear entropy
stability is used to derive provably stable high-order finite-difference methods
with formal boundary closures for conservation laws. Particular emphasis is placed
on the entropy stability of the compressible Navier-Stokes equations. A newly derived
entropy stable weighted essentially non-oscillatory finite-difference method is
used to simulate problems with shocks and a conservative, entropy stable, narrowstencil
finite-difference approach is used to approximate viscous terms.
We present cosmological hydrodynamical simulations of eight Milky Way-sized
haloes that have been previously studied with dark matter only in the Aquarius
project. For the first time, we employ the moving-mesh code AREPO in zoom
simulations combined with a new comprehensive model for galaxy formation
physics designed for large cosmological simulations. Our simulations form in
most of the eight haloes strongly disc-dominated systems with realistic
rotation curves, close to exponential surface density profiles, a stellar-mass
to halo-mass ratio that matches expectations from abundance matching
techniques, and galaxy sizes and ages consistent with expectations from large
galaxy surveys in the local Universe. There is no evidence for any dark matter
core formation in our simulations, even so they include repeated baryonic
outflows by supernova-driven winds and black hole quasar feedback. The
simulations significantly improve upon the results obtained for the same
objects in some of the earlier work based on the SPH technique, and also on the
results obtained in the recent `Aquila' code comparison project which focused
on one of the haloes from our set. For this Aquila object, we carried out a
resolution study with our techniques, covering a dynamic range of 64 in mass
resolution. Without any change in our feedback parameters, the final galaxy
properties are reassuringly similar, in contrast to other modeling techniques
used in the field that are inherently resolution dependent. This success in
producing realistic disc galaxies is reached without resorting to a high
density threshold for star formation, a low star formation efficiency, or early
stellar feedback, factors deemed crucial for disc formation by other recent
numerical studies.
We consider the isentropic compressible Euler system in 2 space dimensions
with pressure law and we show the existence of classical
Riemann data, i.e. pure jump discontinuities across a line, for which there are
infinitely many admissible bounded weak solutions (bounded away from the void).
We also show that some of these Riemann data are generated by a 1-dimensional
compression wave: our theorem leads therefore to Lipschitz initial data for
which there are infinitely many global bounded admissible weak solutions.
Boundary-conforming coordinate transformations are used widely to map a flow region onto a computational space in which a finite-difference solution to the differential flow conservation laws is carried out. This method entails difficulties with maintenance of global conservation and with computation of the local volume element under time-dependent mappings that result form boundary motion. To improve the method, a differential ″geometric conservation law″ (GCL) is formulated that governs the spatial volume element under an arbitrary mapping. Numerical results are presented for implicit solutions of the unsteady Navier-Stokes equations and for explicit solutions of the steady supersonic flow equations.
We study the entropy stability of difference approximations to nonlinear hyperbolic conservation laws, and related time-dependent problems governed by additional dissipative and dispersive forcing terms. We employ a comparison principle as the main tool for entropy stability analysis, comparing the entropy production of a given scheme against properly chosen entropy-conservative schemes.
To this end, we introduce general families of entropy-conservative schemes, interesting in their own right. The present treatment of such schemes extends our earlier recipe for construction of entropy-conservative schemes, introduced in Tadmor (1987 b ). The new families of entropy-conservative schemes offer two main advantages, namely, (i) their numerical fluxes admit an explicit, closed-form expression, and (ii) by a proper choice of their path of integration in phase space, we can distinguish between different families of waves within the same computational cell; in particular, entropy stability can be enforced on rarefactions while keeping the sharp resolution of shock discontinuities.
A comparison with the numerical viscosities associated with entropy-conservative schemes provides a useful framework for the construction and analysis of entropy-stable schemes. We employ this framework for a detailed study of entropy stability for a host of first- and second-order accurate schemes. The comparison approach yields a precise characterization of the entropy stability of semi-discrete schemes for both scalar problems and systems of equations.
We extend these results to fully discrete schemes. Here, spatial entropy dissipation is balanced by the entropy production due to time discretization with a suffciently small time-step, satisfying a suitable CFL condition. Finally, we revisit the question of entropy stability for fully discrete schemes using a different approach based on homotopy arguments. We prove entropy stability under optimal CFL conditions.
In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations.
This chapter discusses a few results on the theoretical side of conservation laws concerning the convergence of approximate solutions. The general setting is a system of n conservation laws in one space dimension. In the context of conservation laws, the maximum norm and the total variation norm provide a natural pair of metrics in which to investigate stability; the maximum norm serving as a measure of the amplitude of the solution and the total variation norm as a measure of the gradient of the solution. Their role is indicated by the theorem concerning the stability and convergence of Glimm's random choice method applied to the Cauchy problem with initial data having small total variation.
A numerical method for simulations of flow over variable geometries including deformable domains or moving boundaries is presented in the context of high-order discontinuous Galerkin (DG) methods in the framework of an arbitrary Lagrangian Eulerian (ALE) description to take into account the deformable domains where boundaries are moving with prescribed motions or deformed under external forces. The ALE DG approach is shown to satisfy the geometric conservation law while preserving high-order accuracy. For variable geometries, we develop a simple and effective mesh smoothing technique based on element size functions to handle deformable grids due to moving boundaries. The current approach is applied to simulations of moving domain problems including laminar flows over oscillating cylinders and a flapping foil to show the ability of handling variable geometries with large deformation and high accuracy. The simulation results are compared with earlier experimental and numerical studies.
Numerical simulations of flow problems with moving boundaries commonly require the solution of the fluid equations on unstructured and deformable dynamic meshes. In this paper, we present a unified theory for deriving Geometric Conservation Laws (GCLs) for such problems. We consider several popular discretization methods for the spatial approximation of the flow equations including the Arbitrary Lagrangian-Eulerian (ALE) finite volume and finite element schemes, and space-time stabilized finite element formulations. We show that, except for the case of the space-time discretization method, the GCLs impose important constraints on the algorithms employed for time-integrating the semi-discrete equations governing the fluid and dynamic mesh motions. We address the impact of these constraints on the solution of coupled aeroelastic problems, and highlight the importance of the GCLs with an illustration of their effect on the computation of the transient aeroelastic response of a flat panel in transonic flow.
The objective of this paper is to establish a firm theoretical basis for the enforcement of discrete geometric conservation laws (D-GCLs) while solving flow problems with moving meshes. The GCL condition governs the geometric parameters of a given numerical solution method, and requires that these be computed so that the numerical procedure reproduces exactly a constant solution. In this paper, we show that this requirement corresponds to a time-accuracy condition. More specifically, we prove that satisfying an appropriate D-GCL is a sufficient condition for a numerical scheme to be at least first-order time-accurate on moving meshes.
Discrete geometric conservation laws (DGCLs) govern the geometric parameters of numerical schemes designed for the solution of unsteady flow problems on moving grids. A DGCL requires that these geometric parameters, which include among others grid positions and velocities, be computed so that the corresponding numerical scheme reproduces exactly a constant solution. Sometimes, this requirement affects the intrinsic design of an arbitrary Lagrangian Eulerian (ALE) solution method. In this paper, we show for sample ALE schemes that satisfying the corresponding DGCL is a necessary and sufficient condition for a numerical scheme to preserve the nonlinear stability of its fixed grid counterpart. We also highlight the impact of this theoretical result on practical applications of computational fluid dynamics.
The formulation and implementation of higher-order accurate temporal schemes for dynamic unstructured mesh problems which satisfy the discrete conservation law are presented. The general approach consists of writing the spatially-discretized equations for an arbitrary-Lagrange–Eulerian system (ALE) as a non-homogeneous coupled set of ODE’s where the dependent variables consist of the product of the flow variables with the control volume. Standard application of backwards difference (BDF) and implicit Runge–Kutta (IRK) schemes to these ODE’s, when grid coordinates and velocities are known smooth functions of time, results in the design temporal accuracy of these schemes. However, in general, these schemes do not satisfy the GCL and are therefore not conservative. Using a suitable approximation of the grid velocities evaluated at the locations in time prescribed by the specific ODE time integrator, a GCL compliant scheme can be constructed which retains the design temporal accuracy of the underlying ODE time integrator. This constitutes a practical approach, since the grid velocities are seldom known as continuous functions in time. Numerical examples demonstrating design accuracy and conservation are given for one, two, and three-dimensional inviscid flow problems.
The derivation of low-storage, explicit Runge–Kutta (ERK) schemes has been performed in the context of integrating the compressible Navier–Stokes equations via direct numerical simulation. Optimization of ERK methods is done across the broad range of properties, such as stability and accuracy efficiency, linear and nonlinear stability, error control reliability, step change stability, and dissipation/dispersion accuracy, subject to varying degrees of memory economization. Following van der Houwen and Wray, sixteen ERK pairs are presented using from two to five registers of memory per equation, per grid point and having accuracies from third- to fifth-order. Methods have been tested with not only DETEST, but also with the 1D wave equation. Two of the methods have been applied to the DNS of a compressible jet as well as methane-air and hydrogen-air flames. Derived 3(2) and 4(3) pairs are competitive with existing full-storage methods. Although a substantial efficiency penalty accompanies use of two- and three-register, fifth-order methods, the best contemporary full-storage methods can be nearly matched while still saving 2–3 registers of memory.
We present a matrix-free discontinuous Galerkin method for simulating compressible viscous flows in two- and three-dimensional moving domains. To this end, we solve the Navier–Stokes equations in an arbitrary Lagrangian Eulerian (ALE) framework. Spatial discretization is based on standard structured and unstructured grids but using an orthogonal hierarchical spectral basis. The method is third-order accurate in time and converges exponentially fast in space for smooth solutions. A novelty of the method is the use of a force-directed algorithm from graph theory that requires no matrix inversion to efficiently update the grid while minimizing distortions. We present several simulations using the new method, including validation with published results from a pitching airfoil, and new results for flow past a three-dimensional wing subject to large flapping insect-like motion.
This paper reviews the symmetrizability of systems of conservation laws which possess entropy functions. Symmetric formulations in conservation form for the equations of gas dynamics are presented.
We present a new reconnection-based arbitrary-Lagrangian–Eulerian (ALE) method. The main elements in a standard ALE simulation are an explicit Lagrangian phase in which the solution and grid are updated, a rezoning phase in which a new grid is defined, and a remapping phase in which the Lagrangian solution is transferred (conservatively interpolated) onto the new grid. In standard ALE methods the new mesh from the rezone phase is obtained by moving grid nodes without changing connectivity of the mesh. Such rezone strategy has its limitation due to the fixed topology of the mesh. In our new method we allow connectivity of the mesh to change in rezone phase, which leads to general polygonal mesh and allows to follow Lagrangian features of the mesh much better than for standard ALE methods. Rezone strategy with reconnection is based on using Voronoi tessellation. We demonstrate performance of our new method on series of numerical examples and show it superiority in comparison with standard ALE methods without reconnection.
We describe a method for computing time-dependent solutions to the compressible Navier–Stokes equations on variable geometries. We introduce a continuous mapping between a fixed reference configuration and the time-varying domain. By writing the Navier–Stokes equations as a conservation law for the independent variables in the reference configuration, the complexity introduced by variable geometry is reduced to solving a transformed conservation law in a fixed reference configuration. The spatial discretization is carried out using the Discontinuous Galerkin method on unstructured meshes of triangles, while the time integration is performed using an explicit Runge–Kutta method. For general domain changes, the standard scheme fails to preserve exactly the free-stream solution which leads to some accuracy degradation, especially for low order approximations. This situation is remedied by adding an additional equation for the time evolution of the transformation Jacobian to the original conservation law and correcting for the accumulated metric integration errors. A number of results are shown to illustrate the flexibility of the approach to handle high-order approximations on complex geometries.
A local theory of weak solutions of first-order nonlinear systems of conservation laws is presented. In the systems considered, two of the characteristic speeds become complex for some achieved values of the dependent variable. The transonic “small disturbance” equation is an example of this class of systems. Some familiar concepts from the purely hyperbolic case are extended to such systems of mixed type, including genuine nonlinearity, classification of shocks into distinct fields and entropy inequalities. However, the associated entropy functions are not everywhere locally convex, shock and characteristic speeds are not bounded in the usual sense, and closed loops and disjoint segments are possible in the set of states which can be connected to a given state by a shock. With various assumptions, we show (1) that the state on one side of a shock plus the shock speed determine the state on the other side uniquely, as in the hyperbolic case; (2) that the “small disturbance” equation is a local model for a class of such systems; and (3) that entropy inequalities and/or the existence of viscous profiles can still be used to select the “physically relevant” weak solution of such a system.
We study how to approximate the metric terms that arise in the discontinuous spectral element (DSEM) approximation of hyperbolic
systems of conservation laws when the element boundaries are curved. We first show that the metric terms can be written in
three forms: the usual cross product and two curl forms. The first curl form is identical to the “conservative” form presented
by Thomas and Lombard [(1979), AIAA J. 17(10), 1030–1037]. The second is a coordinate invariant form. We prove that in two space dimensions, the typical approximation
of the cross product form does satisfy a discrete set of metric identities if the boundaries are isoparametric and the quadrature
is sufficiently precise. We show that in three dimensions, this cross product form does not satisfy the metric identities,
except in exceptional circumstances. Finally, we present approximations of the curl forms of the metric terms that satisfy
the discrete metric identities. Two examples are presented to illustrate how the evaluation of the metric terms affects the
satisfaction of the discrete metric identities, one in two space dimensions and the other in three.