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This paper presents a robust and efficient Poisson solver that can produce accurate solutions and gradi- ents (e.g., heat flux) on unstructured tetrahedral grids. The solver is constructed based on the hyperbolic method for diffusion, where the Laplacian operator is discretized in the form of a hyperbolic system with solution gradients introduced as additional unknown variables. A practical formula for defining a reference length is proposed, which is needed to properly scale the relaxation length associated with the hyper- bolic formulation for scale-invariant computations. The resulting system of residual equations is efficiently solved by a Jacobian-Free Newton-Krylov solver with an implicit defect-correction solver used as a variable- preconditioner. Robustness and superior gradient accuracy are demonstrated for linear and nonlinear Poisson equations through a series of numerical tests with unstructured tetrahedral grids.
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Preprint accepted in Journal of Computational Physics (originally submitted in September, 2019).
Download the journal version at https://doi.org/10.1016/j.jcp.2020.109358
A Hyperbolic Poisson Solver for Tetrahedral Grids
Hiroaki Nishikawa
National Institute of Aerospace, Hampton, VA 23666, USA
March 23, 2020
Abstract
This paper presents a robust and ecient Poisson solver that can produce accurate solutions and gradi-
ents (e.g., heat flux) on unstructured tetrahedral grids. The solver is constructed based on the hyperbolic
method for diusion, where the Laplacian operator is discretized in the form of a hyperbolic system with
solution gradients introduced as additional unknown variables. A practical formula for defining a reference
length is proposed, which is needed to properly scale the relaxation length associated with the hyper-
bolic formulation for scale-invariant computations. The resulting system of residual equations is eciently
solved by a Jacobian-Free Newton-Krylov solver with an implicit defect-correction solver used as a variable-
preconditioner. Robustness and superior gradient accuracy are demonstrated for linear and nonlinear Poisson
equations through a series of numerical tests with unstructured tetrahedral grids.
1 Introduction
The Poisson equation:
div(νgrad u)=f(x, y, z),(1)
where uis a solution variable (e.g., temperature), νis a positive coecient (e.g., heat conductivity), and
fis a forcing term, arises across many disciplines of science and engineering: e.g., the pressure equation in
incompressible flows [1], steady heat conduction [2], electrostatic problems [3], static-state elasticity problems [4],
geophysical electromagnetic modeling [5], and so on. Whether it is linear with νas a function of space (including
a constant) or nonlinear with νas a function of u, ecient numerical methods are available for Cartesian grids
[6, 7, 8]. But current methods for unstructured grids are far from optimal in eciency, robustness, and accuracy.
The demand for ecient, robust, and accurate unstructured-grid solvers, however, only continues to grow as
computational methods are being applied to many practical problems with increasingly complex geometries.
Since there is no systematic ordering in an unstructured grid, e.g., no grid lines along the coordinate directions (or
any particular direction), special solution algorithms such as a Fourier transform or Thomas’ algorithm cannot be
applied. Therefore, unstructured-grid solvers need to rely on iterative methods. Classical iterative methods such
as the Jacobi and Gauss-Seidel relaxation schemes can be easily implemented but are known to significantly
slow down for large-scale grids [9, 10]. Multigrid methods, which in principle can achieve grid-independent
convergence [10], are still not widely employed in practical solvers because their implementation is not simple and
requires further developments for fully unstructured grids. Moreover, robustness issues (e.g., solver divergence)
often arise in iterative methods especially for highly irregular grids. Such issues are often resolved by repairing
the grid but, in general, should be dealt with by improving discretizations and solvers because grids can and will
be highly irregular in adaptive-grid simulations [11]. For these reasons, the development of robust unstructured-
grid solvers remains an active area for research [12, 13, 14, 15]. In addition, it is well known that gradient
accuracy severely deteriorates on unstructured grids [16, 17, 18]. This problem is highly relevant to the Poisson
equation since many quantities governed by the Poisson equation carry physical importance in their gradients
such as heat flux, pressure gradient, stresses, etc., and gradient prediction is often the primary target for
numerical simulations. Despite progress made over the past years [17, 19, 20], satisfactory techniques are not
available yet and a grid with some regularity needs to be generated for each practical simulation. In this
paper, we address these issues of iterative convergence and accuracy by the hyperbolic method for diusion
and demonstrate that a practical Poisson solver can be constructed, which achieves robust convergence and
Associate Research Fellow (hiro@nianet.org), 100 Exploration Way, Hampton, VA 23666 USA,
1
accurate gradient prediction on fully irregular tetrahedral grids. Although not discussed in this paper, the
developed hyperbolic solver will be directly applicable to three-dimensional unsteady diusion problems with
implicit time-stepping schemes [21].
The hyperbolic method for diusion is a method for discretizing a second-order-derivative operator, e.g.,
the left hand side of Eq.(1), in the form of a hyperbolic system, introducing solution gradients as additional
variables [22]:
τu=xp+yq+zrf,
τp=ν
Tr
(xup/ν),
τq=ν
Tr
(yuq/ν),(2)
τr=ν
Tr
(zur/ν),
where τis a pseudo time variable, Tris a relaxation time, and p,q, and rare called the gradient variables. Note
that the above system is equivalent to the Poisson equation (1)with(p, q, r)=νgradu, for any nonzero Tr,
in the pseudo steady state or as soon as the pseudo time derivatives are dropped. Therefore, one can derive a
consistent discretization for the Poisson equation by discretizing the above system by a method for hyperbolic
systems (e.g., an upwind method) and then dropping the pseudo time derivatives. Although the hyperbolic
system (2) looks similar to the classical hyperbolic heat equations of Refs.[23, 24], it is fundamentally dierent
in that Tris a free parameter and can be defined to improve iterative convergence. This is the core idea of the
hyperbolic method considered here. The relaxation time is then defined by
Tr=L2
r
ν,(3)
where Lris a relaxation length, and an optimal formula for Lris given by
Lr=1
2π,(4)
which was derived by maximizing the eect of error propagation with a Fourier analysis on a unit square domain
[22, 25].
It should be noted also that the hyperbolic method considered here is not, unlike the classical approach
[23, 24, 26, 27], proposed a new physical model for diusion and Poisson equations; it is merely a way of con-
structing a superior spatial discretization for the second-derivative diusion operator expressed in a consistent
first-order system form. As it has been demonstrated for many problems since Ref.[22], upwind discretizations
of the hyperbolized diusion system (2) have unique and superior properties over conventional discretizations:
improved order of accuracy and higher quality in the solution gradients on unstructured grids due to the strong
coupling among the solution and gradient variables, and convergence acceleration by the elimination of second
derivatives. Improvements in iterative convergence, especially for implicit methods, come also from enabling
a first-order viscous Jacobian, which is more accurate than a typical zeroth order Jacobian used in conven-
tional viscous solvers [28, 29, 30]. These advantages have been successfully demonstrated for linear/nonlinear
anisotropic diusion equations [31], a tensor-coecient diusion equation [32, 33], incompressible/compressible
Navier-Stokes equations [18, 34, 35, 36], an incompressible magnetohydrodynamics model [37], quasi-neutral
plasma models [32, 38], and the dispersion equation by Ricchiuto’s non-standard hyperbolic formulation [39].
There is, however, a long-standing issue of scale-invariant computations in three-dimensional problems, which
has prevented the method from being applied to practical problems and dimensional equations. For example, if
uis temperature in a heat conduction problem with ν= 1, f= 0, and specified boundary values (i.e., Dirichlet
conditions), the temperature distribution in a given domain must not depend on the unit used to define the grid
(e.g., meter, inch, foot, etc.). This property is referred to, in this paper, as the scale-invariance property. It is
very important to preserve this property in the discrete level because the grid unit should be totally arbitrary
(i.e., freely chosen by a user) and thus it should not greatly aect the numerical solution and cause any serious
problem in numerical simulations. Figure 1 shows an example from Ref.[21], where numerical solutions of the
Laplace equation with Dirichlet conditions obtained by a scale-invariant method on the same triangular grid
with three dierent units, i.e., meter, kilometer, and millimeter, are shown. Note that the solution variation
2
along the boundary (i.e., the same Dirichlet conditions) is kept the same for all grids in order to simulate a
situation where the same physical problem is solved with dierent units. As shown in Figure 1(d), an implicit
solver based on a non-scale-invariant method is very sensitive to the grid unit and even diverges for the grid
defined in millimeter. On the other hand, as in Figure 1(e), a solver based on a scale-invariant method converges
in the same way at the same number of iterations and produces the same numerical solution for all grid units
[21].
x
y
00.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75
(a) Grid[m].
x
y
00.0005 0.001
0
0.0005
0.001
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75
(b) Grid[km]
x
y
0200 400 600 800 1000
0
200
400
600
800
1000
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75
(c) Grid[mm]
Iteration
Max L1 norm of the residual components
Linear relaxation / Iteration
010 20 30 40 50 60
10-10
10-8
10-6
10-4
10-2
100
102
0
100
200
300
400
500
[km]
[m]
[mm]
[km]
[m]
[mm]
(d) Not scale-invariant.
Iteration
Max L1 norm of the residual components
Linear relaxation / Iteration
010 20 30 40 50 60
10-10
10-8
10-6
10-4
10-2
100
0
100
200
300
400
500
[km]
[m]
[mm]
[km]
[m]
[mm]
(e) Scale invariant.
Figure 1: Solution (e.g., temperature) contours obtained by a scale-invariant method in a square domain in three
dierent units (top) and iterative convergence histories for a non-scale-invariant solver and a scale-invariant
solver (bottom) [21].
Many numerical schemes directly discretizing the second-order operator preserves the scale-invariance prop-
erty, but those based on a first-order system formulation, e.g., the hyperbolic method considered here and related
methods such as the one in Ref.[26], do not automatically satisfy this property and can encounter diculties
in iterative convergence although numerical schemes are consistent and maintain the design order of accuracy.
This problem was first recognized when the hyperbolic method was applied to a dimensional diusion equation
and addressed in Ref.[21] for two-dimensional problems. As discussed in Ref.[21], successful scale-invariant
computations require two items: (1) a reference length Lto scale Lrin a consistent manner with the grid unit
in the form:
Lr=L
2π,(5)
(or equivalently to nondimensionalize the hyperbolic system in a consistent manner) and (2) an optimal value of
Lr/L for fast convergence, which ensures, for example, that all Fourier modes propagate just like advection. The
3
first item eliminates the dependence on the grid unit and the second ensures optimal performance of numerical
schemes. To take into account the two, we proposed in Ref.[21], as a practical technique, to define Lsuch
that Lr/L =1/(2π) is optimal in a rectangular domain and then generalize the resulting formula to arbitrary
domains. The proposed formula was demonstrated for various two-dimensional diusion problems, but it was
not immediately applicable to three-dimensional problems. In this paper, we derive an optimal formula for a
three-dimensional rectangular domain, carefully extend it to a general domain, and demonstrate scale-invariance
and superior properties of the hyperbolic solver in three dimensions. The proposed formula thus paves the way
for the hyperbolic method to be applied for solving numerous practical applications.
As the hyperbolic method being a reformulation technique, the hyperbolized Poisson system can be dis-
cretized by any discretization method. In this paper, we employ the node-centered edge-based discretization,
which is one of the most ecient discretization methods and has widely been used in practical unstructured-grid
fluid-dynamics solvers [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. In particular, we consider the edge-based
discretization on fully unstructured tetrahedral grids. Among various types of elements (e.g., hexahedra, prisms,
etc.), we focus on tetrahedral elements for three reasons. First, the edge-based discretization is formally second-
order accurate on arbitrary tetrahedral grids while it deteriorates to first-order with other types elements unless
the grid is regular (see Ref.[52] and Appendix B in Ref.[53]). To maintain accuracy for problems with complex
geometries, it is therefore necessary to develop a method for purely tetrahedral grids. Second, ecient and accu-
rate practical computations will eventually involve grid adaptation, if not used routinely today, and tetrahedral
elements are known to be more flexible and suitable for anisotropic grid adaptation [54, 55, 56, 57, 58, 59, 60].
Finally, although not pursued in this paper, a very economical third-order method can be constructed for the
edge-based method without increasing the number of quadrature points and also without high-order curved
surface mesh on tetrahedral grids [61, 62, 63, 64]. This special third-order property holds only on tetrahedral
grids.
To solve the resulting discrete equations, we first consider an implicit defect-correction (IDC) solver and then
construct a Jacobian-Free Newton-Krylov (JFNK) solver with the IDC solver used as a variable-preconditioner
[13, 21]. The IDC solver is constructed with an approximate residual Jacobian based on the derivative of a
compact lower-order residual; the Jacobian of the second-order residual can be too large to store (and hard to
relax) and thus is not practical. Therefore, the IDC solver is constructed within the same framework for both
linear and nonlinear Poisson equations. JFNK solvers, which solve the exactly linearized system without storing
the true Jacobian, have been increasingly popular in recent years for its robustness especially on unstructured
grids [15, 65]. Its robustness and eciency will be demonstrated here for solving linear and nonlinear Poisson
equations on unstructured tetrahedral grids. However, for the hyperbolic Poisson discretization, the JFNK
solver is not scale-invariant, unless carefully constructed, because it treats all variables and residual equations
with mixed units as a single vector of variables. A suitable modification is proposed in this paper, with which
the scale-invariance property can be guaranteed with the JFNK solver.
The paper is organized as follows. In Section 2, we describe the hyperbolic Poisson system, discuss the
eigen-structure, and derive a dissipation matrix, which will be used to construct an upwind numerical flux. In
Section 3, we describe the edge-based discretization of the hyperbolic Poisson system and boundary conditions.
In Section 4, we discuss how the discretization is aected by the grid unit, derive an optimal reference length for
a rectangular domain, and generalize it to arbitrary domains. In Section 5, we describe two solvers used in the
study: IDC and JFNK solvers, and a modification necessary in the JFNK solver for ensuring the scale-invariance
property. In Section 6, we present numerical results for model problems of heat-conduction in bounded domains,
which is one of our primary target applications. Finally, we conclude the paper with remarks.
2 Hyperbolic Formulation of Poisson Equation
Consider the hyperbolic formulation of the Poisson equation written in the preconditioned conservative form:
P1τU+xF+yG+zH=S,(6)
where
P1=
10 0 0
0Tr/ν00
00Tr/ν0
00 0Tr/ν
,U=
u
p
q
r
,F=
p
u
0
0
,G=
q
0
u
0
,H=
r
0
0
u
,S=
f
p/ν
q/ν
r/ν
.(7)
4
The gradient variables correspond to the fluxes when the pseudo time derivative terms are dropped:
p=ν∂xu, q =ν∂yu, r =ν∂zu, (8)
and therefore the solution gradient can be obtained by
gradu=(xu, yu, zu)='p
ν,q
ν,r
ν(.(9)
This particular formulation is called the nonlinear formulation and suitable for nonlinear diusion problems,
where νis a function of the solution (also useful when νis a function of space) [66, 67]. As shown below, this
formulation allows us to derive the flux Jacobian without dierentiating νfor nonlinear equations.
For the preconditioned conservative formulation, a matrix relevant to the construction of the dissipation
matrix for an upwind numerical flux is given by (see, e.g., Ref.[68])
P1|PAn|,(10)
where
An=Fn
U=(Fˆnx+Gˆny+Hˆnz)
U,(11)
and ˆ
n=(ˆnx,ˆny,ˆnz) is an arbitrary unit vector. Note that the flux vector Fndoes not involve the coecient ν
and therefore there is no need to dierentiate ν. To construct the matrix (10), we first obtain
An=
0ˆnxˆnyˆnz
ˆnx000
ˆny000
ˆnz000
,(12)
and then
PAn=Fn
U=
0ˆnxˆnyˆnz
νˆnx/Tr000
νˆny/Tr000
νˆnz/Tr000
.(13)
The eigenvalues of PA nare given by
±)ν/Tr,0,0,(14)
which are used to define the diagonal matrix:
Λ=λ
1000
0100
0000
0000
,(15)
where
λ=)ν/Tr=ν/Lr.(16)
The corresponding right eigenvectors are given by
R=
Lr/νLr/ν00
ˆnxˆnxˆnyˆnz
ˆnyˆnyˆnx0
ˆnzˆnznx
.(17)
Finally, the dissipation matrix is obtained as
P1|PAn|=P1R|Λ|R1=
ν/Lr0 0 0
0Lrˆn2
x/νLrˆnxˆny/νLrˆnxˆnz/ν
0Lrˆnyˆnx/νLrˆn2
y/νLrˆnyˆnz/ν
0Lrˆnzˆnx/νLrˆnzˆny/νLrˆn2
z/ν
.(18)
5
3 Edge-Based Discretization
3.1 Residual
Given a tetrahedral grid, we store the solution values at nodes and discretize the system (6) at a node jas
VjP1
j
dUj
dτ=*
k{kj}
ΦjkAjk +SjVj,(19)
where Vjis the measure of the dual control volume around the node j,P1
jis the preconditioning matrix
evaluated at jin case that νis not a constant, {kj}is a set of neighbor nodes of j,Ajk is the magnitude of
the directed area vector njk, which is a sum of the directed-areas corresponding to the dual-triangular faces
associated with all tetrahedral elements sharing the edge [j, k] (see Figure 2), and Φjk is a numerical flux
evaluated by the following upwind flux:
Φjk(UL,UR,ˆ
njk)= 1
2[Fn(UR)+Fn(UL)] 1
2P1|PAn|(URUL),(20)
where the projected flux Fnand the Jacobian Ancomputed with the unit directed-area vector: ˆ
njk =njk/|njk|.
Note that the volume Vjis precisely equal to 1/4 of the sum of the volumes of the tetrahedra sharing the node
j. The left and right states, ULand UR, are computed, for second-order accuracy, by the kappa reconstruction
scheme [69, 70]:
UL=Uj++1κ
2(jkUjUjk)+ 1+κ
2Ujk,,(21)
UR=Uk+1κ
2(jkUkUjk)+ 1+κ
2Ujk,,(22)
where κis a real-valued parameter, and
jkUj=ULSQ
j·(xkxj),
jkUk=ULSQ
k·(xkxj),Ujk =1
2(UkUj).(23)
The nodal gradients ULSQ
jand ULSQ
kare computed by a weighted least-squares (LSQ) method:
uLSQ
j=
ˆ
xuj
ˆ
yuj
ˆ
zuj
=*
k{kj}
cx
jk
cy
jk
cz
jk
(ukuj),(24)
where (cx
jk,c
y
jk,c
z
jk) are LSQ coecients (i.e., the coecients of ukujin the solution of a linear LSQ problem),
which are obtained by solving the overdetermined problem (e.g., by the QR factorization):
Ax =b,(25)
where
A=
w1(x1xj)w1(y1yj)w1(z1zj)
.
.
..
.
.
wk(xkxj)wk(ykyj)wk(zkzj)
.
.
..
.
.
wnj(xnjxj)wnj(ynjyj)wnj(znjzj)
,x=
xuj
yuj
zuj
,b=
w1(u1uj)
.
.
.
wk(ukuj)
.
.
.
wnj(unjuj)
,(26)
wk=1
dP
k
,d
k=-(xkxj)2+(ykyj)2+(zkzj)2,(27)
where njis the number of neighbor nodes in {kj}and Pis a parameter, which can take zero (unweighted LSQ),
one (fully weighted LSQ), or any other real value but set to be 0.5 in this study for the hyperbolic Poisson
6
solver. The resulting scheme is called Scheme-I [25] and it is known to bes a very robust scheme. There is a
more ecient scheme called Scheme-II, where the LSQ gradients of the primal solution variable uare replaced
by the gradient variables:
jkuj='p
ν,q
ν,r
ν(j·(xkxj),
jkuk='p
ν,q
ν,r
ν(k·(xkxj).(28)
This scheme does not require LSQ gradients for the primal variables and is typically more accurate. For Scheme-
II, the choice of the parameter κhas been found to be critical for robustness. A typical choice κ= 0 works fine
for isotropic grids, but can lead to instability for anisotropic grids. This issue can be avoided by defining κas
κ=.0ifARjk <10,
0.5ifARjk 10,
(29)
where ARjk is a local grid-aspect-ratio defined at the edge [j, k] as
ARjk = max(ARj,ARk),ARj= max
T{Tj}ART,ARk= max
T{Tk}ART,(30)
where {Tj}and {Tk}denote the sets of tetrahedra around the nodes jand k, respectively, and ARTis the
cell-aspect-ratio defined for a tetrahedron Tas the ratio of the maximum edge length to the minimum height
(i.e., 3 ×volume divided by the maximum triangular face area). As can be seen from Equations (21) and (22),
the contribution of the nodal gradients is reduced for κ>0 and eventually vanishes at κ= 1. Therefore, κ=0.5
has the eect of reducing the contribution of ULSQ
jby half in the reconstruction and it provides a favorable
eect when the gradient is inaccurate and will negatively aect the solver stability. Note, however, that κ=1
cannot be taken because it makes the dissipation term identically vanish and the solver can become unstable.
Not used in this work, but it is also possible to use ARjk for other purposes, e.g., to switch from Scheme-II
to Scheme-I when ARjk is exceptionally large, for increased robustness. Note that the value of κthus defined
depends only on the geometry of the grid and therefore is fixed for a given grid. A more sophisticated control
may be devised by the use of limiter functions [71] although it introduces nonlinearity in the algorithm even for
linear problems. In this study, we have found the above control of κvery eective and do not consider limiter
functions.
Finally, we drop the pseudo time derivative term and define the residual equation at the node j:
Resj≡− *
k{kj}
ΦjkAjk +SjVj=0,(31)
which leads to the global system of 4Nresidual equations:
Res(U)=
Res1(1)
Res1(2)
Res1(3)
Res1(4)
Res2(1)
Res2(2)
Res2(3)
Res2(4)
.
.
.
=0,(32)
to be solved for the vector of 4Nunknowns, U=(u1,p
1,q
1,r
1,u
2,p
2,q
2,r
2,···,u
N,p
N,q
N,r
N), where Nis the
total number of nodes in a given grid and Resj=(Resj(1),Res
j(2),Res
j(3),Res
j(4)). The system of residual
equations is eciently solved by implicit methods, which will be discussed later in Section 5. It is possible
to keep the pseudo time derivative, if desired, possibly to increase solver robustness, but such has been found
unnecessary for the test problems considered in this paper and therefore it is completely dropped here. As a
result, the residual equation (31) becomes a consistent approximation to the Poisson equation expressed in the
first-order system, i.e., the system (2) with the left hand side ignored.
Compared with a conventional Poisson solver that directly discretizes Equation (1), the hyperbolic method
requires the solution of a larger discrete system with 3Nadditional equations and unknowns. Therefore,
7
it requires more memory to store the additional quantities and also more computational time per iteration.
However, as already demonstrated for many problems in previous papers [22, 25, 34, 36], iterative methods
converge more rapidly for the hyperbolic discretization than for a conventional discretization with a speed-up
factor O(1/h), where his a typical mesh spacing, because of the elimination of second derivatives from target
governing equations. Moreover, it leads to a superior implicit solver with a more accurate residual Jacobian
(i.e., the derivative of a compact first-order accurate residual) than that for a conventional scheme, where the
Jacobian is typically constructed with a compact zeroth-order accurate residual [28, 29, 30]. Also, as we will
demonstrate, the hyperbolic solver can produce accurate and useful numerical solutions (accurate gradients,
in particular) on unstructured grids when a conventional solver cannot. This is a decisive advantage of the
hyperbolic solver because then eciency comparison does not even make sense as there is no point of obtaining
inaccurate solutions faster. It is critically important, in a practical point of view, to be able to produce useful
solutions and gradients on fully unstructured grids even with a larger memory requirement.
3.2 Boundary Conditions
Boundary conditions are imposed weakly through the right state that is sent to the numerical flux at bound-
ary nodes. The residual equations are, therefore, defined not only at interior nodes but also at boundary
nodes, and numerical solutions will be determined by solving them at all nodes. To guarantee second-order
accuracy, the boundary residual must be closed by boundary fluxes with an accuracy-preserving boundary-
flux quadrature formula, which depends on the type of elements adjacent to the boundary. A comprehensive
list of second-order quadrature formulas for quadrilateral/triangular elements in two dimensions and hexahe-
dral/tetrahedral/prismatic/pyramidal elements in three dimensions can be found in Appendix B in Ref.[53].
For tetrahedral grids, consider a dual face of a control volume around a boundary node jas shown in Figure
3. The residual at the node jneeds to be closed by the flux contribution across the boundary dual face. The
boundary flux contribution should be computed by the following second-order accurate formula [53, 72, 73]:
+6
8Φj(Uj,UR,ˆ
nB)+1
8Φ2(U2,UR,ˆ
nB)+1
8Φ3(U3,UR,ˆ
nB),|nB|,(33)
where nBdenotes the boundary element normal vector (pointing outward from the interior domain) and
ˆ
nB=nB/|nB|with |nB|being 1/3 of the triangle area. The flux at each node is computed by the upwind flux
(20) with the left state ULgiven by the state stored at the boundary node (e.g., UL=Ujat the node j) and
the right state URdefined by the boundary condition. Note that URdepends on ULand therefore is dierent
at nodes j, 2, and 3. For the Poisson equation, two conditions are considered: Dirichlet and Neumann conditions.
Dirichlet condition: The primal solution value is given: u=ub,whereubis a given value. This condi-
tion is implemented as
UR=
uR
pR
qR
rR
=
2ubuL
pL
qL
rL
,(34)
where p,q, and rare evaluated by the left state since their values are not known, so that the average of uLand
uRmatches the given solution:
uL+uR
2=ub.(35)
Neumann condition: The gradient normal to the boundary surface is given: u/n=pnb,wherenis
taken in the direction towards the interior domain and pnbis the given normal gradient. This condition is
implemented as
UR=
uR
pR
qR
rR
=
uL
pL+ 2(pnbpn
Ln
x
qL+ 2(pnbpn
Ln
y
rL+ 2(pnbpn
Ln
z
,(36)
8
where the solution value is copied from the left state since it is not known and pn=(p, q, r)·ˆ
n
Bwith ˆ
n
B=
n
x,ˆn
y,ˆn
z)=ˆ
nB, so that the average of the normal derivatives matches the given value:
pn
L+pn
R
2=pnb.(37)
Note that the Neumann condition specifies the normal derivative at a boundary, but there is no need, in the
hyperbolic method, to discretize the normal derivative at a boundary because gradients are available directly
at boundary nodes as variables and they are second-order accurate. In eect, the Neumann condition can be
imposed in the same way as the Dirichlet condition via gradient variables, which is another useful feature of the
hyperbolic method.
4 Defining Reference Length for Scale Invariance
4.1 Scale Dependence
As previously discussed in Ref.[21] for two-dimensional problems, the relaxation length Lrneeds to be scaled
by a suitable reference length in order to ensure the scale-invariance property: the numerical solution to the
Laplace equation (f= 0 and ν= 1) must not depend on the grid unit. In conventional methods, as one can
easily see, for example, from the classical finite-dierence scheme on a Cartesian grid with a uniform spacing h,
ui1,j,k 2ui,j,k +ui+1,j,k
h2+ui,j1,k 2ui,j,k +ui,j +1,k
h2+ui,j,k12ui,j,k +ui,j,k+1
h2=0,(38)
which is
ui,j,k =1
6[ui1,j,k +ui+1,j,k +ui,j1,k +ui,j+1,k +ui,j,k1+ui,j,k+1],(39)
the discretization and thus the numerical solution do not depend on (x, y, z), and hence, it is scale invariant.
It is pointed out at this point that the scale-invariance property is defined for the Laplace equation. For
a more general equation such as linear and nonlinear Poisson equations, the discretization is said to be scale
invariant if it has the scale-invariance property when applied to the Laplace equation. Scale-invariant discretiza-
tions will be scale invariant for linear and nonlinear Poisson equations if ν(x, y, z , u) and f(x, y, z) are defined
consistently with the grid unit used as we will show later in Section 6 for cases where f=f(x, y, z) and ν=ν(u).
On an unstructured grid, many conventional discretizations of the Poisson equation will depend on (x, y, z )
but they will not depend on the grid unit. For example, consider the residual defined by a conventional scheme
[53, 74]:
Resj=*
k{kj}+'uLSQ
j+uLSQ
k(·ˆ
njk +α
|ˆ
ejk ·ˆ
njk|Ljk
(uRuL),Ajk =0,(40)
where ˆ
ejk =ejk/|ejk|,ejk =(xkxj,y
kyj,z
kzj), Ljk =|ejk|=)(xkxj)2+(ykyj)2+(zkzj)2,uL
and uRare computed by the kappa reconstruction scheme, and the damping parameter αis given an optimal
value of 4/3[53, 74]. This is a generalized version of the classical scheme called the Mathur-Murthy [75, 76]
or face-tangent scheme [29], and has been used in practical unstructured-grid codes [42, 77]. Observe that all
terms in the square bracket is of O(1/length) and therefore their balance is not aected by the change in the
grid unit and the same numerical solution will be obtained for dierent grid units. This makes sense since, for
example in the case of a steady heat conduction problem, a temperature distribution is uniquely determined
for a given shape of the domain and boundary conditions, and it should not depend on the scale of the domain.
However, in the hyperbolic method, the discretization is aected by the coordinate scaling unless a special care
is taken. To see this, consider the first component of the residual (31)withf= 0 and ν= 1:
*
k{kj}+(pL,q
L,r
L)·ˆ
njk +(pR,q
R,r
R)·ˆ
njk +1
Lr
(uRuL),Ajk =0.(41)
The first two terms in the square bracket are solution gradients (e.g., p=xusince ν= 1). Therefore, they
change their values when the grid unit is changed. For example, if uis temperature in Kelvin, [K], then
p= 300[K/m]p= 1000/300[K/mm]. It means that the third term (i.e., dissipation) in the square bracket
9
must also change its magnitude in the same manner. Otherwise, the discrete equation will change by the amount
of dissipation, thus leading to changes in the solution and iterative convergence, which is exactly the case when
Lr=1/(2π). This is the scale-invariance problem originally pointed out in Ref.[21]. Note, however, that the
dierence is only in the amount of dissipation and thus the discretization remains consistent and second-order
accurate. But second-order accurate solutions are not guaranteed to be obtained because the solver may get
unstable and diverge by the altered dissipation. To ensure the scale-invariance property, the relaxation length
must be scaled by a reference length, L:
Lr=L
2π,(42)
where Lneeds to be defined for a given problem and we cannot set L=Ljk as in the conventional scheme
because then the hyperbolic discretization will lose its advantages and reduce to a conventional scheme [22].
This raises two questions: (1)How should we define Lfor a given problem?, and (2)Is Lr/L =1/(2π), which
was derived for a square domain, still optimal for other shapes of domains and what value is optimal if not?
The first question is dicult to answer for the Poisson equation, where there are no particular representative
length scales; or it is easy to answer since it can be any length, which however makes the second question more
dicult because the choice of Laects the optimal value of Lr/L.InRef.[21], we answered these questions, in
two dimensions, by deriving a formula for Lthat makes Lr/L =1/(2π) optimal for a rectangular domain and
then rearranging it in a form that can be applied to arbitrary domains. Below, we extend the approach to three
dimensions and develop a practical formula for L.
Remark: As mentioned in Introduction, many discretization methods for diusion inherently possess the
scale-invariance property. However, those involving a characteristic length of O(1) in the discretization, e.g., a
method in Ref.[78] and a hybridizable discontinuous Galerkin method with a stabilization parameter of O(1)
[79], will need a similar technique to properly choose and/or scale a characteristic length for a given grid in
order for them to behave consistently (i.e., to give the same error and iterative convergence) for dierent grid
units.
4.2 Rectangular Domain
Consider a rectangular domain: (x, y , z)[0,L
x]×[0,L
y]×[0,L
z]. Assume, for simplicity, a Cartesian grid
with a uniform spacing h=Lx/Nx=Ly/Ny=Lz/Nz,whereNx,Ny, and Nzare the numbers of cells in x-,
y, and z-coordinate directions, respectively. Substitute a Fourier mode,
U0exp [i(βxx/h +βyy/h +βzz/h)],(43)
where U0=[u0,p
0,q
0,r
0] is a vector of amplitudes, βx,βy, and βzare frequencies in x,y, and zdirections,
respectively, and i=1, into the first-order version of the residual (i.e., no linear reconstruction, and with
f= 0) to get
dU0
dτ=MU0,(44)
where Mis given for smooth components as
M=
νβ2
2hLr
iβx
h
iβy
h
iβz
h
iν2βx
hL2
rν
L2
r
00
iν2βy
hL2
r
0ν
L2
r
0
iν2βz
hL2
r
00ν
L2
r
,(45)
whose eigenvalues are
ν
L2
r
,ν
L2
r
,ν
2L2
r/1+Lrβ2
2h±1
Lr01Lrβ2
h+β2L2
r
4h2(β216)1.(46)
10
Although slightly dierent from those in Ref.[21] because of the nonlinear formulation employed in this paper,
it is very convenient that the expression inside the square root is exactly the same as in the two-dimensional
case. The relaxation length is determined so as to keep the last two eigenvalues complex conjugate for all
possible frequencies, so that Fourier modes will propagate just like advection instead of being purely damped
(see Ref.[21] for the derivation):
Lr2h
(β+ 4)β,(47)
which we impose with the smooth-smooth-smooth component (βx,β
y,β
z)=(π/Nx,π/N
y,π/N
z)withβ=
-β2
x+β2
y+β2
z=π
˜
N:
Lr=2h
(π/˜
N+ 4)π/˜
N,(48)
where
˜
N=1
-1/N 2
x+1/N 2
y+1/N 2
z
=1
h-1/L2
x+1/L2
y+1/L2
z
.(49)
It is further simplified for 1/˜
N1 as
Lr=1
2πh˜
N=1
2π
1
-1/L2
x+1/L2
y+1/L2
z
,(50)
from which we identify the optimal Lfor a rectangular domain, which is denoted by Lopt, as
Lopt =1
-1/L2
x+1/L2
y+1/L2
z
.(51)
If the domain of interest is indeed rectangular, one can employ this formula to compute Lopt and use it to scale
Lras Lr=Lopt/(2π). However, it cannot be directly applied to arbitrary domains. To generalize it to arbitrary
domains, we consider writing the formula in terms of computable quantities such as the volume and surface
area of the domain as discussed further in the next section.
4.3 Arbitrary Domain
For an arbitrary domain, it is not possible to derive Lopt in the same way because the Fourier analysis is not
applicable. In two dimensions, the optimal reference length Lopt, which can be obtained also from Equation (51)
by taking Lz→∞, can be fully expressed in terms of the area and the perimeter of a rectangular domain. The
resulting expression can be directly applied to arbitrary domains with the area and the perimeter computed from
a given computational grid; and it was indeed found to serve very well as a reference length to guarantee the
scale-invariance property and achieve fast iterative convergence [21]. In three dimensions, we can express Lopt
given in Equation (51) in terms of the volume V=LxLyLz, the total surface area S= 2(LxLy+LyLz+LzLx),
and the diagonal distance Diag2=L2
x+L2
y+L2
z:
Lopt =V
-S2/42V)Diag2+S
.(52)
This formula can be directly applied to arbitrary domains, but it requires the computation of the diagonal
distance (i.e., such as the maximum distance between two boundary points), which may not be as simple and
ecient as one might hope for. As a practical option, we propose the following formula:
Diag2= max(˜
L2
x,˜
L2
y,˜
L2
z),(53)
where
˜
Lx=xmax xmin,˜
Ly=ymax ymin,˜
Lz=zmax zmin,(54)
11
and xmin and xmax denote the minimum and maximum x-coordinates of the nodes in a given grid, and similarly
for others. As one might have noticed, the diagonal formula (53) does not reduce precisely to the diagonal
distance when the domain is rectangular. However, the dierence is very minor and computational results
for a rectangular domain are almost the same whether the above approximation is used or not. Also, there
is another reason that we recommend the approximation (53). When the Lopt formula (52) is applied to an
arbitrary domain, there is no guarantee that the expression inside the square root, i.e., S2/42V)Diag2+S,
is positive. In fact, it can get negative for a sphere if we compute the diagonal distance, for example, by
Diag2=˜
L2
x+˜
L2
y+˜
L2
z, which is a reasonable formula. To see this, write the term inside the square root in
Equation (52) as
S2/42V)Diag2+S=S2+1/42
(S/V ))Diag2/S2+1/S,.(55)
This needs to be positive but can be negative if the domain has too small Sfor a fixed V. The limiting case is
a sphere, which has the smallest surface area for a given volume. Consider a sphere of radius R, for which we
have V=4πR3/3 and S=4πR2. For Diag2=˜
L2
x+˜
L2
y+˜
L2
z= 12R2,wefind
S2/42V)Diag2+S=S2+1/42R
3)12R2/(4πR2)2+1/(4πR2),(56)
=S2+3π43+π
12π,(57)
≈−0.01294770009 ×S2,(58)
but for Equation (53), we have Diag2=4R2and thus
S2/42V)Diag2+S=S2+1/42R
3)4R2/(4πR2)2+1/(4πR2),(59)
=S2+3π41+π
12π,(60)
0.03407020948 ×S2>0.(61)
Therefore, the formula (52) with the diagonal estimate (53) is expected to be valid for arbitrary domains.
To summarize, for a given grid, we first compute
V=*
j{J}
Vj,S=*
E{Eb}
SE,(62)
where {J}is the set of all nodes in the grid, {Eb}is the set of all triangular faces of all the boundaries in the
grid, and SEis the area of the triangle E, then compute Lopt by the formula (52)withDiag2as in Equation
(53), and finally define the relaxation length Lras
Lr=Lopt
2π.(63)
Note that Lopt can be computed conveniently by using nodal coordinates, cell-volumes, cell-face areas avail-
able from a given computational grid. However, it is not a grid-dependent quantity but a domain-dependent
quantity. Of course, if it is known that the domain is rectangular, one can directly employ the optimal formula
given in Equation (51). The generalized formula (52) is proposed for more complex domains, but can also be
used for a rectangular domain without significantly degrading the performance of the hyperbolic solver. It is
emphasized also that Lopt must be computed and Lrmust be defined as above even when the grid coordinates
are already non-dimensionalized in order to eliminate the ambiguity in choosing the reference length used for
non-dimensionalization [21]. Finally, as in Ref.[21], we focus here on problems over bounded domains and leave
discussions on open-boundary problems to future work, which will be discussed in relation to fluid-dynamic
applications. Note, however, that the proposed technique can be directly applied to open-boundary problems
of any kind. It is just not guaranteed to be optimal and may require some minor adjustments for the best per-
formance. Also, for fluid-dynamic applications, one can alternatively nondimensionalize the problem of interest
for fluid dynamic equations as typically done and then set Lto be the length scale used to nondimensionalize
the fluid dynamic equations as it has been done for the hyperbolic Navier-Stokes method [18, 34, 35, 36].
12
5 Solvers
To eciently solve the global system of residual equations, we employ implicit solvers: IDC and JFNK solvers.
The IDC solver is simple and ecient, but encounters diculties on irregular grids. Nevertheless, it can still
serve well as a variable-preconditioner and help the JFNK solver converge robustly and eciently. The JFNK
solver is therefore the recommended solver for practical problems, but it requires a careful implementation
in order to ensure the scale-invariant convergence as discussed below. Note that these solvers are typically
employed for solving nonlinear discrete equations but used here also for linear discrete equations arising from
the discretization of a linear Poisson equation because the matrix of the linear discrete equations can be too
expensive to store.
5.1 Implicit Defect Correction (IDC) Solver
The IDC solver is an iterative method defined by
Um+1 =Um+U,(64)
where U=(u1,p
1,q
1,r
1,u
2,p
2,q
2,r
2,···,u
N,p
N,q
N,r
N) is the global solution vector, mis the iteration
counter, and the correction Uis computed by solving the linearized system:
Res
UU=Res(Um),(65)
where the Jacobian Res/Uis the exact derivative of the first-order version of the residual Res (i.e., with
zero LSQ gradients but the Scheme-II reconstruction (28) retained). It is emphasized that the first-order
Jacobian is an advantage of the hyperbolic solver and an improvement over conventional viscous schemes, for
which a Jacobian based on a zeroth-order scheme is common (because a first-order version does not typically
exist)[28, 29, 30]. To solve the linear system, we employ a multi-color Gauss-Seidel relaxation scheme with a
specified tolerance of one order of magnitude residual reduction or for a specified maximum number (25 or 100) of
relaxations. As previously demonstrated for two-dimensional problems (and will be shown for three-dimensional
problems later), the IDC solver is very ecient and scale invariant: convergence histories for dierent grid units
match if the residual norms are scaled by the initial values. The same is true in three dimensions. Although we
have found that it is not as robust and ecient as one would wish (e.g., the linear relaxation may diverge on
highly distorted grids), it still serves well as a variable-preconditioner for the JFNK solver as described in the
next section.
5.2 Jacobian-Free Newton-Krylov (JFNK) Solver
5.2.1 Standard implementation
The JFNK solver is constructed based on the algorithm presented in Ref.[13], with the generalized conjugate
residual (GCR) method: with x0=0,r0=Res(Um), p0=˜
A1r0, perform for i=0,1,2,··· ,i
max
αi=(Api)Tri
(Api)T(Api),xi+1 =xi+αipi,ri+1 =riαiApi,, (66)
where imax is a user-defined integer, and, if not converged yet, compute pi+1 =˜
A1ri+1, perform the orthogo-
nalization for k=0,1,2,···,i:
βi=(Api)T(Apk)
(Apk)T(Apk),pi+1 =pi+1 +βkpk,Api+1 =Api+1 βkApk,(67)
and go back to the step (66). At convergence, we obtain the correction as U=xkand update the solution
by Equation (64). The symbol ˜
A1denotes the variable-preconditioner based on the multi-color Gauss-Seidel
relaxation, which approximately inverts the approximate Jacobian ˜
A=Res/Uwith the tolerance of 50%
residual reduction or for a specified maximum number 25 of relaxations. Furthermore, A=Res/Uis the
true Jacobian, but the computation and storage are avoided by computing the matrix-vector product, e.g., Apk,
by the Fechet derivative approximation:
Apk=Res(Um+ϵpk)Res(Um)
ϵ,(68)
13
where
ϵ= max(1,|Um|)1016.(69)
5.2.2 Scale-invariant implementation
Unlike the IDC solver, the JFNK solver as described above is not scale-invariant because it treats all
variables/residuals with mixed units as a single vector of solutions/residuals. In our case, the variable udoes
not change its value for a change in the grid unit, but the variables p,q, and rdo change their values. Then,
the length (or the norm) of a vector and dot products will change in a manner that will change the behavior of
the solver. In order to make the solver scale-invariant, we need to unify the units of the solution variables and
residual components. It can be done conveniently with the reference length L, which is used to scale Lr. Define
the 4N×4Ndiagonal matrix:
Ds= diag
21,1
L,1
L,1
L,1,1
L,1
L,1
L,···3.(70)
Then, compute r0as
r0=DsRes(Um),(71)
and re-scale the Jacobian used in the preconditioner as
˜
A=(DsRes)
(D1
sU),(72)
which means that each 4×4 Jacobian block, expressed here by ai,j, is scaled as
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
a11 a12/L a13 /L a14/L
a21/L a22 /L2a23/L2a24 /L2
a31/L a32 /L2a33/L2a34 /L2
a41/L a42 /L2a43/L2a44 /L2
,(73)
and also that the solution xhas just been re-scaled as xD1
sx. Furthermore, the Fechet derivative needs
to be computed as
Apk=Ds
Res(Uk+ϵDspk)Res(Um)
ϵ,(74)
where
ϵ= max(1,|D1
sUm|)1016.(75)
As a result, the correction Uis obtained as
U=Dsxk.(76)
Note that Lshould be uniquely defined for a given problem, and Lopt serves the purpose very well. These
re-scalings will guarantee the dimensional consistency in xand make the algorithm scale-invariant.
6 Results
In this section, we present numerical results and demonstrate the scale-invariance property and advantages
of the hyperbolic Poisson solver for problems with various domain shapes. In the first four problems, we solve
a linear Poisson equation with ν= 1:
div(grad u)=f(x, y, z),(77)
14
|ˆ
ejk ·ˆ
njk|AR
Nodes Tetrahedra Min. Ave. Max. Min. Ave. Max.
Grid1 512 2,058 0.19 0.83 1.0 1.4 3.9 97.3
Grid2 4,096 20,250 0.21 0.84 1.0 1.4 3.3 613.4
Grid3 32,768 178,746 0.20 0.85 1.0 1.3 3.3 41983.8
Grid4 110,592 622,938 0.22 0.85 1.0 1.3 3.0 160.2
Grid5 252,144 1,500,282 0.25 0.86 1.0 1.4 2.9 276.3
Table 1: Summary of grid statistics (independent of the grid unit) for cubic-domain grids. The skewness measure
|ˆ
ejk ·ˆ
njk|is defined at edges, and the aspect ratio AR is defined at cells.
where the forcing term fwill be determined for a chosen exact solution. The IDC solver is used with the
maximum of 100 relaxations and the tolerance of 0.1 for the first two problems, and the maximum of 25
relaxations and the tolerance of 0.5 (i.e., the reduction of the linear-system residual by half) for all other
problems. These values are representative of those typically used in practical applications, so as to move onto
the next iteration without spending too much time with the linear solver. As a preconditioner in the JFNK
solver, the latter set of parameters (i.e., 25 and 0.5) are always used. The JFNK solver uses the maximum 10
Krylov vectors (i.e., imax = 10) and the tolerance of 0.1. For all problems, the initial solution is set to be 1 for
the variable u, and zero for all the gradient variables. The solvers are taken to be converged, unless otherwise
stated, when all the four residual components are reduced in the L1norm by six orders of magnitude from their
initial values. Iterative convergence histories will be shown for the maximum of the L1norms (over the four
equations) scaled by the corresponding initial norm over the four equations. Results will be compared for L=1
and L=Lopt with Lrgiven as in Equation (42). All computations were performed by a serial code; a study on
eects of parallelization is left as future work.
6.1 Cubic domain
We begin with the linear Poisson equation (77) with the following forcing function:
f(x, y, z)=π2(a2+b2+c2)sin(π(ax +by +cz)),(78)
which leads to the following exact solution:
u(x, y, z)=sin(π(ax +by +cz)),(79)
where a=2.2, b=2.3, and c=2.4, to be solved in a cubic domain with the weak Dirichlet condition specified
by the exact solution as described in Section 3.2.
We generated a series of five irregular tetrahedral grids for a cubic domain. As the baseline, the side length
1 is considered as measured in meter, [m]. To demonstrate that the same numerical solution is obtained by
the hyperbolic solver with the optimal reference length Lopt, we solve the same problem with grids expressed
in three dierent units: [km], [m], and [mm]. The coarsest grids are shown in Figure 4 for dierent units,
which are referred to as Grid1[km], Grid1[m], Grid1[mm], respectively, and similarly for finer levels. Note that
the exact solution (79) is always computed with the baseline coordinates (i.e., in [m]), the exact gradients are
computed in the same way with the baseline coordinates but then divided by 0.001 for grids in [km] and 1000
for grids in [mm], and the forcing term (78) is computed again with the baseline coordinates but then will be
divided by 0.0012for grids in [km] and 10002for grids in [mm]. The same applies to all the test problems that
follow. Table 1 shows the grid statistics, which is independent of the grid unit. The skewness measure |ˆ
ejk ·ˆ
njk|
is computed at each edge, where ˆ
ejk is the unit vector of the edge between nodes jand k, and ˆ
njk is the unit
directed-area vector. A very small value indicates high skewness: the two vectors do not align with each other
and make an angle close to 90 degrees. The aspect ratio AR is defined for each cell as the ratio of the longest
edge length to the smallest height computed for the largest triangular face area. As can be seen, these grids
are isotropic with low skewness and cell-aspect-ratio. Grid3 exhibits a very large maximum aspect ratio, but it
is an isolated eect as the average is very low and close to those for other grids. The optimal reference length
15
|ˆ
ejk ·ˆ
njk|AR
Nodes Tetrahedra Min. Ave. Max. Min. Ave. Max.
Grid1 512 2,058 0.00033 0.014 1.0 518 2035 57,958
Grid2 4,096 20,250 0.00030 0.057 1.0 635 1826 441,588
Grid3 32,768 178,746 0.00027 0.038 1.0 572 1825 16,912,443
Grid4 110,592 622,938 0.00040 0.035 1.0 566 1692 81,377
Grid5 252,144 1,500,282 0.00034 0.033 1.0 568 1670 141,358
Table 2: Summary of grid statistics (independent of the grid unit) for high-aspect-ratio rectangular grids. The
skewness measure |ˆ
ejk ·ˆ
njk|is defined at edges, and the aspect ratio AR is defined at cells.
Lopt was computed, as described in Section 4, for each grid, but it is essentially the same for all levels of grids
in the same grid unit and diers only by the magnitude for dierent units: 0.5192793E-03 in [km], 0.5192793
in [m], and 0.5192793E+03 in [mm].
Convergence results for the IDC solver with L= 1 and L=Lopt are shown in Figure 5 for the finest
grid, Grid5[m], Grid5[km], Grid5[mm], where the solid lines indicate the residual histories and the dashed lines
indicate the number of linear relaxations per iteration. Note again that the residual norm is the maximum
among the four equations and normalized by the corresponding initial norm. As can be seen in Figure 5(a),the
solver behaves very dierently, as expected, with L= 1: it converges reasonably fast for Grid5[m], significantly
slows down for Grid5[km], and diverges for Grid5[mm]. This is the problem with L= 1 as reported for two-
dimensional problems in Ref.[21]. On the other hand, convergence histories perfectly match, as desired, with
L=Lopt as shown in Figure 5(b). Superior gradient accuracy by the hyperbolic solver is demonstrated in
Figure 6, where error convergence results for L=Lopt are shown for u,r=zu, and zuLSQ. Here, the errors
in the gradient are multiplied by 0.001 for grids in [km] and 1000 for grids in [mm], so that their magnitude
match across the three units. Clearly, all the error norms match in magnitude, and the solution uand the
gradient rare second-order accurate while the LSQ gradient is only first-order accurate. The same order of
accuracy in the solution and the gradient is one of the advantageous features of the hyperbolic method, and
it is demonstrated here to produce much more accurate gradients than the LSQ gradients computed from the
soluion. Although not shown, other gradients pand qare also obtained with second-order accuracy. Figure 7
provides a qualitative comparison of the z-derivative over the boundary surface for Grid2[m]. As expected, the
contours contain noise for the LSQ gradient (Figure 7(c)) whereas those for the gradient variable rare much
smoother (Figure 7(b)) and very close to those of the exact derivative (Figure 7(a)).
6.2 High-aspect-ratio rectangular domain
To investigate the impact of high skewness and large cell aspect ratio, we created a flat domain from the
previous cubic domain by re-scaling the z-axis by a factor of 1/1000. The coarsest grids are shown in Figure 8,
again for the three dierent grid units. The same forcing and exact solutions as in the previous case were used,
except that we set c= 200.4 in order to generate enough solution variation in the z-direction. As can be seen in
the grid statistics shown in Table 2, these grids are highly skewed with |ˆ
ejk ·ˆ
njk|0.035, which corresponds to
an angle of 89.8, and flat with the aspect ratio of O(1000) on average. Lopt was, again, computed for each grid,
but it is essentially the same for all levels of grids in the same grid unit: 9.997313E-07 in [km], 9.997313E-04 in
[m], and 9.997313E-01 in [mm].
Convergence histories are shown for Grid5[km], Grid5[m], and Grid5[mm] in Figure 9. The solver is not
scale-invariant with L= 1 and significantly slows down for Grid5[km] and Grid5[m]. On the other hand, as
shown in Figure 9(b), the solver converges for all grids in the scale-invariant manner for L=Lopt. However,
for this problem, the solver encounters diculties in the linear relaxation, hitting the maximum of 100 at every
iteration, which can take up significant computational time for fine 3D grids. Then, the JFNK solver was
applied to see if it improves convergence and eciency. Results are shown in Figures 9(c) and 9(d). Figure 9(c)
shows that the JFNK solver converged rapidly in seven iterations and it is scale-invariant as desired. Note that
the linear relaxation in the plots as indicated by “Relax” in the legend is the total number of preconditioner
relaxations performed per iteration. The number of Krylov vectors used per iteration is indicated by “‘KV” in
16
the legend. Figure 9(d) compares the IDC and JFNK solvers in terms of CPU time and clearly indicates that
the JFNK solver converges faster than the IDC solver. These results demonstrate that the JFNK solver can be
more ecient and reliable than the IDC solver for highly stretched grids.
Error convergence results are shown in Figure 10, again, for u,r, and zuLSQ. As before, the hyperbolic
solver achieves second-order accuracy for both the solution and gradients, and the LSQ gradient is first-order
accurate. In Figure 11, contours of zuare compared on Grid3[m]. It is observed again that the LSQ gradient
is very noisy on this grid while the gradient variable r=zuis very smooth and close to the exact contours.
It is important to note that the noise in the LSQ gradient does not vanish on finer grids. This can be seen in
Figure 12, which shows the contours on Grid5[m]. Noise can be seen although smaller than that on Grid3[m].
6.3 Hemisphere cylinder
To explore other shapes of domains, we consider a domain around a hemisphere-cylinder enclosed by a large
hemisphere and a plane at the base. For eciency, we consider only the half of the entire domain (y>0).
See Figure 13(a), which shows the domain without the outer half-hemisphere boundary. For this problem, we
consider solving the linear Poisson equation with the following forcing function:
f(x, y, z)=A(a2+b2+c2)π2sin(aπx) cos(bπy)sin(cπz),(80)
and the exact solution:
u(x, y, z)=Asin(aπx) cos(bπy)sin(cπz),(81)
where A= 100, a=0.05, b=0.025, and c=0.5. The Dirichlet condition is applied at the hemisphere-cylinder
and the outer half-hemisphere; the Neumann condition is imposed elsewhere (i.e., two plane boundaries), where
the solution derivative normal to the boundary is zero. The problem is solved, again, in three dierent units:
[km], [m], and [mm], and with four levels of grids. The coarsest grids are shown for the three dierent units in
Figures 13(b), 13(c), and 13(d). As shown in Table 3, these grids are relatively skewed and high-aspect-ratio
near the hemisphere-cylinder surface with the minimum skewness measure being around 0.01 and the maximum
aspect ratio of O(100). Lopt was computed as 8.725970E-03 in [km], 8.725970 in [m], and 8.725970E+03 in
[mm].
|ˆ
ejk ·ˆ
njk|AR
Nodes Tetrahedra Min. Ave. Max. Min. Ave. Max.
Grid1 680 2,880 0.0097 0.54 1.0 2.2 27.2 206.2
Grid2 4,587 23,040 0.0083 0.53 1.0 2.2 26.5 240.0
Grid3 33,605 184,320 0.0076 0.53 1.0 2.2 26.4 261.7
Grid4 257,097 1,474,560 0.0073 0.53 1.0 2.2 26.4 274.0
Table 3: Summary of grid statistics (independent of the grid unit) for the hemisphere-cylinder grids. The
skewness measure |ˆ
ejk ·ˆ
njk|is defined at edges, and the aspect ratio AR is defined at cells.
For this problem, we consider both the IDC and JFNK solvers. Convergence histories are shown in Figure
14. As expected, the IDC solver is not scale-invariant for L= 1 as shown in Figure 14(a). On the other hand, as
shown in Figure 14(b), it is scale-invariant with L=Lopt. However, the linear relaxation hits the maximum of
25 every time as in the previous case, meaning that 25 relaxations is not enough to reduce the linear residual by
half. It is, nevertheles, sucient as a preconditioner in the JFNK solver. As shown in Figure 14(c). the JFNK
solver converges rapidly in 9 iterations and it is scale-invariant. The 10th iteration in Grids[m] is considered
due to a round-oeect, where the residual norm was actually reduced by nearly six orders of magnitude in the
9th iteration (as can be seen in the figure) but not exactly. Figure 14(d) shows that the JFNK solver is faster
in CPU time than the IDC solver. Although the linear relaxation in the preconditioner hits the maximum of
25 and also it requires 10 Krylov vectors in the last six iterations, the JFNK solver converges very fast in terms
of CPU time.
Figure 15 shows error convergence results. As expected, second-order accuracy is observed for the solution
uand the gradient variable r=zu, but lower-order accuracy (if not first-order) is observed for the LSQ
17
gradient zuLSQ. Not shown, but results are similar in other gradient components. As can be seen in the
comparison given in Figure 16, large LSQ gradient errors occur near the outer hemisphere boundary. These
plots demonstrate, again, the superior gradient accuracy by the hyperbolic solver.
6.4 Curved tube
To further investigate the impact of Lopt for a more general domain, we consider a curved-tube domain as
shown in Figure 17. We solve, again, the linear Poisson equation with the Dirichlet condition. The same forcing
term and exact solution are used, as in the cubic domain case except a=1.5, b=1.5, and c=1.5. Four levels of
grids have been generated with the coarsest grid as shown in Figure 17 for three dierent units: [km], [m], and
[mm]. Grid statistics is shown in Table 4, in terms of which these grids are similar to those in the previous case
with the minimum skewness measure of O(0.01) and the maximum aspect ratio of O(100). Lopt was computed
as 1.471851E-03 in [km], 1.471851 in [m], and 1.471851E+03 in [mm].
|ˆ
ejk ·ˆ
njk|AR
Nodes Tetrahedra Min. Ave. Max. Min. Ave. Max.
Grid1 6,480 30,240 0.015 0.40 1.0 1.7 24.0 187.5
Grid2 46,656 245,376 0.013 0.40 1.0 1.5 24.0 163.2
Grid3 151,632 832,032 0.013 0.40 1.0 1.5 24.0 155.0
Grid4 352,512 1,976,832 0.013 0.40 1.0 1.5 24.0 151.7
Table 4: Summary of grid statistics (independent of the grid unit) for the curved-tube grids. The skewness
measure |ˆ
ejk ·ˆ
njk|is defined at edges, and the aspect ratio AR is defined at cells.
As before, the IDC solver convergence wildly varies for dierent grid units with L= 1 as shown in Figure
18(a). On the other hand, scale-invariant convergence is achieved with the optimal length L=Lopt as can be
clearly seen in Figure 18(b). Convergence is further improved by the JFNK solver as shown in Figure 18(c):
the JFNK solver is scale-invariant and converges with much less iterations than the IDC solver. It is faster also
in CPU time than the IDC solver as can be seen in Figure 18(d).
Figure 19 shows the error convergence results for u,r=zu, and the LSQ gradient zuLSQ. Again, the
hyperbolic solver gives second-order accuracy not only for ubut also for the gradient r. The LSQ gradient
is significantly less accurate although it seems to exhibit nearly second-order convergence for these grids. In
Figure 20, derivative contours are compared over the inner boundary in Grid2[m], where the grid is stretched
and relatively high-aspect-ratio in the surface normal direction. On this grid, the gradient ris already accurate
and close to the exact contours, but the LSQ gradient contours are distorted and far from the exact contours.
6.5 Nonlinear problem
To demonstrate the capability for nonlinear problems, we consider the following nonlinear Poisson equation:
div(ν(u) grad u)=f(x, y, z),(82)
where
ν(u)=1+u2,f(x, y, z)=2A3(cπ)2cos(cπx) cos(cπy)4{cos(cπx)}2+{cos(cπy)}25exp '32cπz(,(83)
with c=0.5 and A=0.1. The exact solution is given by
u(x, y, z)=Acos(cπx) cos(cπy)exp'2cπz(.(84)
The nonlinear Poisson equation is solved in an incomplete-donut-shaped domain as in Figure 21.Atthetwo
annulus end-planes at x= 0 and y= 0, the Neumann condition is imposed with a zero normal derivative,
and the Dirichlet condition is imposed elsewhere. Three levels of tetrahedral grids were generated by a two-
dimensional irregular triangular grid and then subdividing prisms into tetrahedra; the coarsest grids are shown
18
|ˆ
ejk ·ˆ
njk|AR
Nodes Tetrahedra Min. Ave. Max. Min. Ave. Max.
Grid1 23,328 120,960 0.027 0.49 1.0 3.7 19.9 69.5
Grid2 176,256 981,504 0.011 0.40 1.0 3.8 39.6 165.3
Grid3 1,368,576 7,907,328 0.106 0.40 1.0 3.5 38.9 169.0
Table 5: Summary of grid statistics (independent of the grid unit) for the nonlinear problem. The skewness
measure |ˆ
ejk ·ˆ
njk|is defined at edges, and the aspect ratio AR is defined at cells.
for dierent units in Figure 21. The irregularity of the two-dimensional section grid can be observed in Figure
21(d). These grids are relatively skewed and high-aspect-ratio as in the previous two cases. See Table 5 for grid
statistics. Lopt was computed as 2.685715E-03 in [km], 2.685715 in [m], and 2.685715E+03 in [mm].
For this problem, we compare the hyperbolic solver with a conventional solver, which is constructed based
on the alpha-damping diusion scheme (40) modified for the nonlinear equation as
Resj=*
k{kj}+6ν(uj)uLSQ
j+ν(uk)uLSQ
k7·ˆ
njk +αν(ujk )
|ˆ
ejk ·ˆ
njk|Ljk
(uRuL),Ajk f(xj,y
j,z
j)Vj=0,(85)
where ujk =(uL+uR)/2withuLand uRcomputed by the kappa reconstruction (κ= 0). This is a scalar
residual equation at a node jfor the Poisson equation (82). The same IDC and JFNK solvers are used but they
are applied to the reduced number of residual equations (i.e., Nequations instead of 4N). As the conventional
discretization is scale-invariant with both the IDC and JFNK solvers, we solve the problem only on grids with
the unit [m]. It should be noted also that the Jacobian required in the IDC solver/preconditioner is constructed
by dierentiating the damping term only (i.e., α
|ˆ
ejk·ˆ
njk|Ljk (ukuj)), which by itself is an inconsistent scheme
on skewed grids, and such inconsistent Jacobians are known to degrade iterative convergence of the IDC solver
[28, 29, 30]. In this problem, the tolerance is reduced further and the solver is taken to be converged when the
residual is reduced by eight (not six) orders of magnitude. This is because we observed significant dierences in
the discretization errors for the conventional scheme with six orders of residual reduction; the tolerance of eight
orders of magnitude reduction has been found sucient to obtain converged numerical solutions. Below, in the
figures, the conventional method will be referred to as Alpha4/3.
As in the linear cases, the IDC solver is highly scale-dependent with L= 1 as can be seen in Figure 22(a),
where convergence histories are shown for the finest grid. It is then made completely scale-invariant with
L=Lopt as shown in Figure 22(b). Also plotted here is the convergence history of the conventional solver.
The linear relaxation hits the maximum 25 at almost every iteration, failing to reduce the linear residual; and
thus the solver took more than 150 iterations to converge. As discussed earlier, there are two weak points in
the conventional solver: (1)the Jacobian is constructed as the derivative of an inconsistent scheme (i.e., the
damping term only), which is especially far from the true Jacobian for skewed grids, and (2)a higher condition
number of the linear system by the discretized second-derivative operator. The hyperbolic discretization oers
advantages in both aspects. First, the Jacobian is constructed by the derivative of the first-order accurate
(not inconsistent) scheme. It is therefore closer to the true Jacobian, and thus faster nonlinear convergence is
expected. Second, the condition number is O(1/h), where his a representative mesh spacing, for the hyperbolic
discretization, which is O(h) times smaller than O(1/h2) typical in conventional discretizations. The improved
condition number leads to faster convergence in the linear relaxation, and that is exactly what is observed in
the result shown in Figure 22(b).
More robust and ecient convergence can be achieved for both hyperbolic and conventional methods by the
JFNK solver with the IDC solver used as a preconditioner as shown in Figures 22(c) and 22(d). Figure 22(c)
confirms that the JFNK solver is again scale-invariant and converged with only 8 iterations. It converged rapidly
also for the conventional discretization, but took more Krylov vectors as well as preconditioner relaxations as
can be seen in Figure 22(d), which compares the IDC and JFNK solvers for the hyperbolic discretization and the
JFNK solver for the conventional discretization. Observe the larger numbers of Krylov vectors and relaxations
for the conventional discretization. Notice also that the JFNK solver is significantly faster in CPU time than
the IDC solver.
It may be possible to further reduce the CPU time for the conventional solver, e.g., by smart programming,
19
but such is not even worth considering because the numerical solution obtained with the conventional method
is very inaccurate. Figure 23 shows the error convergence results, again for u,r=zu, and the LSQ gradients
zuLSQ , which are computed for both solutions (i.e., produced with the hyperbolic and conventional methods).
Second-order accuracy is observed for all, but the errors in the LSQ gradients are much larger than the error
in the gradient variable r. Also, the error in uis larger for the conventional method than for the hyperbolic
method. In Figure 24, contours of the derivative normal to the inner surface nuare compared on all grids:
the exact nu,nu=(p, q, r)·n
B, and nu=zuLSQ ·n
Bfor the conventional method. As can be seen, the
contours of nu=(p, q, r)·n
Bis very smooth, accurate, and very close to the exact contours on all grids. On the
other hand, the contours of the normal derivative computed with the LSQ gradient in the conventional method
are very noisy and far from the exact contours. It seems to have improved on the finest grid, but it still contains
noise and is far from the exact contours. Such inaccuracy of gradients on unstructured tetrahedral grids is
very well known. Typically, unstructured tetrahedral grids are avoided for problems where gradient accuracy
is important, and prismatic/hexahedral grids are used instead (if possible). However, as mentioned earlier,
tetrahedral grids are more suitable for automatic grid generation and anisotropic adaptation. The numerical
results shown here indicate that the hyperbolic solver has a potential for making grid generation much easier
for complex geometries and allowing anisotropic grid adaptation methods to be applied to a wider range of
practical problems with purely tetrahedral grids.
6.6 Nonlinear problem over a more complex domain
To demonstrate the solver for a more complex geometry, we consider a spiral-shaped tube as shown in Figure
25 and solve the same nonlinear Poisson equation with c=0.06 and A=0.2 and the same boundary conditions.
Four levels of tetrahedral grids were generated with 23,040, 165,888, 1,253,376, and 9,732,096 nodes; Figure
25 shows the coarsest one. As can be seen from the grid statistics in Table 6, the cell aspect ratio is slightly
smaller than that of the grids used in the previous case. For this problem, we only compare the hyperbolic and
conventional methods with the JFNK solver on grids with a single grid unit. Lopt was computed as 2.563783 for
the finest grid, and it is very similar on other grids. Convergence results are shown for the finest grid in Figure
26. The solver converges very rapidly in 8 iterations for both discretization methods. However, it requires more
linear relaxations and Krylov vectors with the conventional method. As a result, the solver took less CPU time
with the hyperbolic discretization as can be seen in Figure 26(b). Figure 27 shows error convergence results for
uand its y-derivative. The error in uis slightly lower with the conventional method, but the gradient yuis
more accurate with the hyperbolic method. Again, the hyperbolic method generates qualitatively more accurate
gradients as illustrated here for the surface normal gradient plotted over the inner boundary. See Figure 28.
|ˆ
ejk ·ˆ
njk|AR
Nodes Tetrahedra Min. Ave. Max. Min. Ave. Max.
Grid1 23,040 109,728 0.024 0.48 1.0 2.7 20.4 86.4
Grid2 165,888 881,280 0.029 0.49 1.0 2.5 19.0 82.4
Grid3 1,253,376 7,064,064 0.031 0.49 1.0 2.4 18.5 80.8
Grid4 9,732,096 56,567,808 0.019 0.49 1.0 2.1 18.3 84.2
Table 6: Summary of grid statistics (independent of the grid unit) for the second nonlinear test case. The
skewness measure |ˆ
ejk ·ˆ
njk|is defined at edges, and the aspect ratio AR is defined at cells.
7 Concluding Remarks
We have presented a hyperbolic Poisson solver for arbitrary tetrahedral grids, which is fundamentally more
ecient and accurate than conventional methods, delivering second-order accuracy for both the solution and
the gradients on irregular tetrahedral grids. A practical formula has been proposed for estimating a reference
length in three dimensions, which is needed to scale the relaxation length associated with a hyperbolic formu-
lation of the Poisson equation for ensuring the scale-invariance property (i.e., the same solution is obtained
independently of the grid unit). The properly scaled hyperbolic Poisson system is discretized by the edge-based
20
method, and a robust Jacobian-Free Newton-Krylov solver has been developed to solve the resulting system of
residual equations. The developed hyperbolic Poisson solver has been shown to be more robust and accurate
than conventional solvers: (1) faster and more robust iterative convergence by the eect of eliminating second
derivatives from the algorithm (i.e., reduced condition number) and by the use of a more accurate first-order
Jacobian, and (2) superior gradient accuracy, which is not only one order higher than least-squares gradients
but also free from noise even on irregular tetrahedral grids. These advantages have been demonstrated for both
linear and nonlinear Poisson equations over various shapes of domains with unstructured tetrahedral grids. By
eectively resolving the issue of scale-invariance for three-dimensional problems, the work presented has paved
the way for the hyperbolic method to be applied for solving numerous practical applications governed by linear
and nonlinear Poisson equations.
Future work includes computations with adaptive tetrahedral grids, applications to unsteady diusion prob-
lems (e.g., heat transfer, elasticity, etc.) with the developed solver applied within each time step to solve
unsteady residual equations of an implicit time-stepping scheme [21], extensions to other discretization meth-
ods, e.g., cell-centered finite-volume methods, an extension to third-order accuracy based on the third-order
edge-based method [64], and the development of grid-independent multigrid solvers for the preconditioner as
well as for the nonlinear solver, in which the JFNK solver can serve as a robust smoother [10, 80]. Finally,
although the proposed reference-length formula has been shown to be useful for solving linear/nonlinear Poisson
equations over complex domains, it is theoretically optimal only for a rectangular domain and a truly optimal
formula for an arbitrary domain, in terms of both iterative convergence and accuracy, remains to be discovered.
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Acknowledgments
The author gratefully acknowledges support from Software CRADLE, part of Hexagon, and support from
the U.S. Army Research Oce under the contract/grant number W911NF19-1-0429 with Dr. Matthew Munson
as the program manager.
24
j
k
c
m
cR
cL
(a) Dual face contributions from an adjacent tetrahedral
element to the edge [j, k].
j
k
(b) Total dual face at the edge [j, k].
Figure 2: Dual face contribution at the edge [j, k]. A numerical flux is evaluated at the midpoint of the edge
m, indicated by an open circle. The centroid of the tetrahedral element is denoted by c, and the centroids of
the two adjacent triangles are denoted by cLand cR.
j, b
2
3
nB
||
||| |||
Figure 3: A dual-face (shaded area) of a control volume around a boundary node j. The boundary
dual-face normal vector nBis pointing outward from the interior domain.
25
(a) Grid1[km] (b) Grid1[m] (c) Grid1[mm]
Figure 4: The coarsest grids used for the cubic domain test case with dierent grid units.
(a) L=1. (b) L=Lopt .
Figure 5: Comparison of convergence histories for Grid5 in the cubic domain test case.
26
-5 -4.5 -4
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
Figure 6: Error convergence results for u,r=zu, and LSQ =zuLSQ in the cubic domain test case.
(a) Exact zu(b) r=zu(c) zuLSQ
Figure 7: Comparison of zucontours for Grid2[m] in the cubic domain test case.
27
(a) Grid1[km] (b) Grid1[m] (c) Grid1[mm]
Figure 8: The coarsest grids used for the high-aspect-ratio rectangular domain test case with dierent grid
units.
(a) IDC: L=1. (b) IDC: L=Lopt .
(c) JFNK: L=Lopt. (d) IDC and JFNK: L=Lopt.
Figure 9: Comparison of convergence histories for Grid5 in the high-aspect-ratio rectangular domain test case.
28
-6 -5.5 -5
-4
-3
-2
-1
0
1
2
3
Figure 10: Error convergence results for u,r=zu, and LSQ =zuLS Q in the high-aspect-ratio rectan-
gular domain test case.
(a) Exact zu(b) r=zu(c) zuLSQ
Figure 11: Comparison of zucontours for Grid3[m] in the high-aspect-ratio rectangular domain test case.
(a) Exact zu(b) r=zu(c) zuLSQ
Figure 12: Comparison of zucontours for Grid5[m] in the high-aspect-ratio rectangular domain test case.
29
(a) Grid1. (b) Grid1[km]
(c) Grid1[m] (d) Grid1[mm]
Figure 13: The coarsest grids used for the hemisphere-cylinder domain test case with dierent grid units.
30
(a) IDC: L=1. (b) IDC: L=Lopt .
(c) JFNK: L=Lopt. (d) IDC and JFNK: L=Lopt.
Figure 14: Comparison of convergence histories for Grid4 in the hemisphere-cylinder domain test case.
31
-4 -3.5 -3
-1
-0.5
0
0.5
1
1.5
Figure 15: Error convergence results for u,r=zu, and LSQ =zuLS Q in the hemisphere-cylinder
domain test case.
(a) Exact xu(b) p=xu(c) xuLSQ
Figure 16: Comparison of xucontours for Grid4[m] in the hemisphere-cylinder domain test case.
32
(a) Grid1[km] (b) Grid1[m] (c) Grid1[mm]
Figure 17: The coarsest grids used for the curved-tube domain test case with dierent grid units.
(a) IDC: L=1. (b) IDC: L=Lopt .
(c) JFNK: L=Lopt. (d) IDC and JFNK: L=Lopt.
Figure 18: Comparison of convergence histories for Grid4 in the curved-tube domain test case.
33
-4 -3.5 -3
-2.5
-2
-1.5
-1
-0.5
0
0.5
Figure 19: Error convergence results for u,r=zu, and LSQ =zuLSQ in the curved-tube domain test
case.
(a) Exact zu(b) r=zu(c) zuLSQ
Figure 20: Comparison of zucontours for Grid2[m] in the curved-tube domain test case. The outer boundary
is not shown.
34
(a) Grid1[km] (b) Grid1[m]
(c) Grid1[mm](d) Zoomed-in view.
Figure 21: The coarsest grids used for the nonlinear test case with dierent grid units.
35
(a) IDC: L=1. (b) IDC: L=Lopt .
(c) JFNK: L=Lopt. (d) IDC and JFNK: L=Lopt.
Figure 22: Comparison of convergence histories for Grid3[m] in the nonlinear test case.
36
-4 -3.8 -3.6 -3.4
-5
-4
-3
-2
-1
Figure 23: Error convergence results for u,r=zu, and LSQ =zuLSQ in the nonlinear test case.
37
(a) Exact (Grid1). (b) Hyperbolic (Grid1). (c) Alpha4/3 (Grid1).
(d) Exact (Grid2). (e) Hyperbolic (Grid2). (f ) Alpha4/3 (Grid2).
(g) Exact (Grid3). (h) Hyperbolic (Grid3). (i) Alpha4/3 (Grid3).
Figure 24: Comparison of contours of the solution gradient normal to the inner boundary surface nufor
Grid3[m] in the nonlinear test case. The outer boundary is not shown. The normal derivative is computed with
nu=(p, q, r )·n
Bfor the hyperbolic method and nu=uLS Q ·n
Bfor the conventional method (Alpha4/3).
38
Figure 25: The coarsest grid used for the second nonlinear test case.
39
(a) JFNK solver: maximum residual norm versus itera-
tion.
(b) JFNK solver: maximum residual norm versus CPU
time.
Figure 26: Comparison of convergence histories for the hyperbolic and conventional methods on the finest grid
in the second nonlinear test case.
-1 -0.5 0
-5.5
-5
-4.5
-4
-3.5
-3
-2.5
Figure 27: Error convergence results for u,q=yuand
yuLSQ in the second nonlinear test case.
(a) Exact (Grid4). (b) Hyperbolic (Grid4). (c) Alpha4/3 (Grid4).
Figure 28: Comparison of contours of the solution gradient normal to the inner boundary surface nuon
the finest grid in the second nonlinear test case. The outer boundary is not shown. The normal derivative
is computed with nu=(p, q , r)·n
Bfor the hyperbolic method and nu=uLS Q ·n
Bfor the conventional
method (Alpha4/3).
40
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