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1.
Michael Dumbser Ariunaa Uuriintsetseg & Olindo Zanotti 《Communications In Computational Physics》2013,14(2):301-327
In this article we present a new family of high order accurate Arbitrary
Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff
hyperbolic balance laws. High order accuracy in space is obtained with a standard
WENO reconstruction algorithm and high order in time is obtained using the local
space-time discontinuous Galerkin method recently proposed in [20]. In the Lagrangian
framework considered here, the local space-time DG predictor is based on a weak
formulation of the governing PDE on a moving space-time element. For the space-time basis and test functions we use Lagrange interpolation polynomials defined by
tensor-product Gauss-Legendre quadrature points. The moving space-time elements
are mapped to a reference element using an isoparametric approach, i.e. the space-time mapping is defined by the same basis functions as the weak solution of the PDE.
We show some computational examples in one space-dimension for non-stiff and for
stiff balance laws, in particular for the Euler equations of compressible gas dynamics,
for the resistive relativistic MHD equations, and for the relativistic radiation hydrodynamics equations. Numerical convergence results are presented for the stiff case up to
sixth order of accuracy in space and time and for the non-stiff case up to eighth order
of accuracy in space and time. 相似文献
2.
On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes
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Rui Zhang Mengping Zhang & Chi-Wang Shu 《Communications In Computational Physics》2011,9(3):807-827
In this paper we consider two commonly used classes of finite volume
weighted essentially non-oscillatory (WENO) schemes in two dimensional Cartesian
meshes. We compare them in terms of accuracy, performance for smooth and shocked
solutions, and efficiency in CPU timing. For linear systems both schemes are high
order accurate, however for nonlinear systems, analysis and numerical simulation results
verify that one of them (Class A) is only second order accurate, while the other
(Class B) is high order accurate. The WENO scheme in Class A is easier to implement
and costs less than that in Class B. Numerical experiments indicate that the resolution
for shocked problems is often comparable for schemes in both classes for the same
building blocks and meshes, despite of the difference in their formal order of accuracy.
The results in this paper may give some guidance in the application of high order finite
volume schemes for simulating shocked flows. 相似文献
3.
A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws
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Rapha& euml l Loub& egrave re Michael Dumbser & Steven Diot 《Communications In Computational Physics》2014,16(3):718-763
In this paper, we investigate the coupling of the Multi-dimensional Optimal
Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER)
approach in order to design a new high order accurate, robust and computationally
efficient Finite Volume (FV) scheme dedicated to solving nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and
three space dimensions, respectively. The Multi-dimensional Optimal Order Detection
(MOOD) method for 2D and 3D geometries has been introduced in a recent series of
papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite
Volume scheme in space, using polynomial reconstructions with a posteriori detection
and polynomial degree decrementing processes to deal with shock waves and other
discontinuities. In the following work, the time discretization is performed with an
elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say, the optimal high-order of accuracy is reached
on smooth solutions, while spurious oscillations near singularities are prevented. The
ADER technique not only reduces the cost of the overall scheme as shown
on a set of numerical tests in 2D and 3D, but also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO
schemes and the new ADER-MOOD approach has been carried out for high-order
schemes in space and time in terms of cost, robustness, accuracy and efficiency. The
main finding of this paper is that the combination of ADER with MOOD generally
outperforms the one of ADER and WENO either because at given accuracy MOOD isless expensive (memory and/or CPU time), or because it is more accurate for a given
grid resolution. A large suite of classical numerical test problems has been solved
on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations
of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations
(RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic
partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements. 相似文献
4.
Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes
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Walter Boscheri Michael Dumbser & Elena Gaburro 《Communications In Computational Physics》2022,32(1):259-298
We propose a new high order accurate nodal discontinuous Galerkin (DG)
method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical
polynomials of degree $N$ inside each element, in our new approach the discrete solution
is represented by piecewise continuous polynomials of degree $N$ within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon.
We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG
method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each
sub-triangle are defined, as usual, on a universal reference element, hence allowing to
compute universal mass, flux and stiffness matrices for the subgrid triangles once and
for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured
nature of the computational grid. High order of accuracy in time is achieved thanks
to the ADER approach, making use of an element-local space-time Galerkin finite element predictor.The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results
have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained
by the corresponding modal DG version of the scheme. 相似文献
5.
Runge-Kutta Discontinuous Galerkin Method Using WENO-Type Limiters: Three-Dimensional Unstructured Meshes
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This paper further considers weighted essentially non-oscillatory (WENO) and Hermite weighted essentially non-oscillatory (HWENO) finite volume methods as limiters for Runge-Kutta discontinuous Galerkin (RKDG) methods to solve problems involving nonlinear hyperbolic conservation laws. The application discussed here is the solution of 3-D problems on unstructured meshes. Our numerical tests again demonstrate this is a robust and high order limiting procedure, which simultaneously achieves high order accuracy and sharp non-oscillatory shock transitions. 相似文献
6.
Shuangzhang Tu Gordon W. Skelton & Qing Pang 《Communications In Computational Physics》2011,9(2):441-480
This paper presents a novel high-order space-time method for hyperbolic
conservation laws. Two important concepts, the staggered space-time mesh of the
space-time conservation element/solution element (CE/SE) method and the local discontinuous
basis functions of the space-time discontinuous Galerkin (DG) finite element
method, are the two key ingredients of the new scheme. The staggered space-time
mesh is constructed using the cell-vertex structure of the underlying spatial mesh.
The universal definitions of CEs and SEs are independent of the underlying spatial
mesh and thus suitable for arbitrarily unstructured meshes. The solution within each
physical time step is updated alternately at the cell level and the vertex level. For
this solution updating strategy and the DG ingredient, the new scheme here is termed
as the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracy
is achieved by employing high-order Taylor polynomials as the basis functions
inside each SE. The present DG-CVS exhibits many advantageous features such as
Riemann-solver-free, high-order accuracy, point-implicitness, compactness, and ease
of handling boundary conditions. Several numerical tests including the scalar advection
equations and compressible Euler equations will demonstrate the performance of
the new method. 相似文献
7.
A New Approach of High Order Well-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms
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Hyperbolic balance laws have steady state solutions in which the flux gradients are
nonzero but are exactly balanced by the source terms. In our earlier work [31–33], we designed
high order well-balanced schemes to a class of hyperbolic systems with separable source terms.
In this paper, we present a different approach to the same purpose: designing high order
well-balanced finite volume weighted essentially non-oscillatory (WENO) schemes and RungeKutta
discontinuous Galerkin (RKDG) finite element methods. We make the observation that
the traditional RKDG methods are capable of maintaining certain steady states exactly, if a
small modification on either the initial condition or the flux is provided. The computational
cost to obtain such a well balanced RKDG method is basically the same as the traditional
RKDG method. The same idea can be applied to the finite volume WENO schemes. We
will first describe the algorithms and prove the well balanced property for the shallow water
equations, and then show that the result can be generalized to a class of other balance laws.
We perform extensive one and two dimensional simulations to verify the properties of these
schemes such as the exact preservation of the balance laws for certain steady state solutions,
the non-oscillatory property for general solutions with discontinuities, and the genuine high
order accuracy in smooth regions. 相似文献
8.
Jun Zhu & Jianxian Qiu 《Communications In Computational Physics》2020,27(3):897-920
In this paper, a new type of third-order and fourth-order weighted essentially non-oscillatory (WENO) schemes is designed for simulating the Hamilton-Jacobi equations on triangular meshes. We design such schemes with the use of the nodal information defined on five unequal-sized spatial stencils, the application of monotone Hamiltonians as a building block, the artificial set of positive linear weights to make up high-order approximations in smooth regions simultaneously avoiding spurious oscillations nearby discontinuities of the derivatives of the solutions. The spatial reconstructions are convex combinations of the derivatives of a modified cubic/quartic polynomial defined on a big spatial stencil and four quadratic polynomials defined on small spatial stencils, and a third-order TVD Runge-Kutta method is used for the time discretization. The main advantages of these WENO schemes are their efficiency, simplicity, and can be easily implemented to higher dimensional unstructured meshes. Extensive numerical tests are performed to illustrate the good performance of such new WENO schemes. 相似文献
9.
High-Order Runge-Kutta Discontinuous Galerkin Methods with a New Type of Multi-Resolution WENO Limiters on Tetrahedral Meshes
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In this paper, the second-order and third-order Runge-Kutta discontinuous
Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory
(WENO) limiters are proposed on tetrahedral meshes. The multi-resolution WENO
limiter is an extension of a finite volume multi-resolution WENO scheme developed
in [81], which serves as a limiter for RKDG methods on tetrahedral meshes. This new
WENO limiter uses information of the DG solution essentially only within the troubled cell itself which is identified by a new modified version of the original KXRCF
indicator [42], to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the second or third degree of the DG solution. The second-order and
third-order RKDG methods with the associated multi-resolution WENO limiters are
developed as examples for general high-order RKDG methods, which could maintain
the original order of accuracy in smooth regions and keep essentially non-oscillatory
property near strong discontinuities by gradually degrading from the optimal order
to the first order. The linear weights inside the procedure of the new multi-resolution
WENO limiters can be set as any positive numbers on the condition that they sum
to one. A series of polynomials of different degrees within the troubled cell itself
are applied in a WENO fashion to modify the DG solutions in the troubled cell on
tetrahedral meshes. These new WENO limiters are very simple to construct, and can
be easily implemented to arbitrary high-order accuracy on tetrahedral meshes. Such
spatial reconstruction methodology improves the robustness in the simulation on the
same compact spatial stencil of the original DG methods on tetrahedral meshes. Extensive one-dimensional (run as three-dimensional problems on tetrahedral meshes)
and three-dimensional tests are performed to demonstrate the good performance of
the RKDG methods with new multi-resolution WENO limiters. 相似文献
10.
Conservative Semi-Lagrangian Finite Difference WENO Formulations with Applications to the Vlasov Equation
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In this paper, we propose a new conservative semi-Lagrangian (SL) finite
difference (FD) WENO scheme for linear advection equations, which can serve as a
base scheme for the Vlasov equation by Strang splitting [4]. The reconstruction procedure
in the proposed SL FD scheme is the same as the one used in the SL finite volume
(FV) WENO scheme [3]. However, instead of inputting cell averages and approximate
the integral form of the equation in a FV scheme, we input point values and approximate
the differential form of equation in a FD spirit, yet retaining very high order
(fifth order in our experiment) spatial accuracy. The advantage of using point values,
rather than cell averages, is to avoid the second order spatial error, due to the shearing
in velocity (v) and electrical field (E) over a cell when performing the Strang splitting
to the Vlasov equation. As a result, the proposed scheme has very high spatial accuracy,
compared with second order spatial accuracy for Strang split SL FV scheme for
solving the Vlasov-Poisson (VP) system. We perform numerical experiments on linear
advection, rigid body rotation problem; and on the Landau damping and two-stream
instabilities by solving the VP system. For comparison, we also apply (1) the conservative
SL FD WENO scheme, proposed in [22] for incompressible advection problem, (2)
the conservative SL FD WENO scheme proposed in [21] and (3) the non-conservative
version of the SL FD WENO scheme in [3] to the same test problems. The performances
of different schemes are compared by the error table, solution resolution of sharp interface,
and by tracking the conservation of physical norms, energies and entropies,
which should be physically preserved. 相似文献
11.
High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes
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Yizhou Lu Jun Zhu Shengzhu Cui Zhenming Wang Linlin Tian & Ning Zhao 《Communications In Computational Physics》2023,33(5):1217-1239
In this paper, a new multi-resolution weighted essentially non-oscillatory
(MR-WENO) limiter for high-order local discontinuous Galerkin (LDG) method is designed for solving Navier-Stokes equations on triangular meshes. This MR-WENO
limiter is a new extension of the finite volume MR-WENO schemes. Such new limiter
uses information of the LDG solution essentially only within the troubled cell itself, to
build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the
highest degree of the LDG method. As an example, a third-order LDG method with associated same order MR-WENO limiter has been developed in this paper, which could
maintain the original order of accuracy in smooth regions and could simultaneously
suppress spurious oscillations near strong shocks or contact discontinuities. The linear weights of such new MR-WENO limiter can be any positive numbers on condition
that their summation is one. This is the first time that a series of different degree polynomials within the troubled cell are applied in a WENO-type fashion to modify the
freedom of degrees of the LDG solutions in the troubled cell. This MR-WENO limiter
is very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured meshes. Such spatial reconstruction
methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original LDG methods on triangular meshes. Some classicalviscous examples are given to show the good performance of this third-order LDG
method with associated MR-WENO limiter. 相似文献
12.
This paper is concerned with a new version of the Osher-Solomon Riemann
solver and is based on a numerical integration of the path-dependent dissipation matrix.
The resulting scheme is much simpler than the original one and is applicable to
general hyperbolic conservation laws, while retaining the attractive features of the original
solver: the method is entropy-satisfying, differentiable and complete in the sense
that it attributes a different numerical viscosity to each characteristic field, in particular
to the intermediate ones, since the full eigenstructure of the underlying hyperbolic system
is used. To illustrate the potential of the proposed scheme we show applications
to the following hyperbolic conservation laws: Euler equations of compressible gasdynamics
with ideal gas and real gas equation of state, classical and relativistic MHD
equations as well as the equations of nonlinear elasticity. To the knowledge of the authors,
apart from the Euler equations with ideal gas, an Osher-type scheme has never
been devised before for any of these complicated PDE systems. Since our new general
Riemann solver can be directly used as a building block of high order finite volume
and discontinuous Galerkin schemes we also show the extension to higher order of
accuracy and multiple space dimensions in the new framework of PNPM schemes on
unstructured meshes recently proposed in [9]. 相似文献
13.
14.
Comparison of Fifth-Order WENO Scheme and Finite Volume WENO-Gas-Kinetic Scheme for Inviscid and Viscous Flow Simulation
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The development of high-order schemes has been mostly concentrated on
the limiters and high-order reconstruction techniques. In this paper, the effect of the
flux functions on the performance of high-order schemes will be studied. Based on the
same WENO reconstruction, two schemes with different flux functions, i.e., the fifth-order WENO method and the WENO-Gas-Kinetic scheme (WENO-GKS), will be compared. The fifth-order finite difference WENO-SW scheme is a characteristic variable
reconstruction based method which uses the Steger-Warming flux splitting for inviscid terms, the sixth-order central difference for viscous terms, and three stages Runge-Kutta time stepping for the time integration. On the other hand, the finite volume
WENO-GKS is a conservative variable reconstruction based method with the same
WENO reconstruction. But it evaluates a time dependent gas distribution function
along a cell interface, and updates the flow variables inside each control volume by
integrating the flux function along the boundary of the control volume in both space
and time. In order to validate the robustness and accuracy of the schemes, both methods are tested under a wide range of flow conditions: vortex propagation, Mach 3
step problem, and the cavity flow at Reynolds number 3200. Our study shows that
both WENO-SW and WENO-GKS yield quantitatively similar results and agree with
each other very well provided a sufficient grid resolution is used. With the reduction of mesh points, the WENO-GKS behaves to have less numerical dissipation and
present more accurate solutions than those from the WENO-SW in all test cases. For
the Navier-Stokes equations, since the WENO-GKS couples inviscid and viscous terms
in a single flux evaluation, and the WENO-SW uses an operator splitting technique, it
appears that the WENO-SW is more sensitive to the WENO reconstruction and boundary treatment. In terms of efficiency, the finite volume WENO-GKS is about 4 times
slower than the finite difference WENO-SW in two dimensional simulations. The current study clearly shows that besides high-order reconstruction, an accurate gas evolution model or flux function in a high-order scheme is also important in the capturing of physical solutions. In a physical flow, the transport, stress deformation, heat conduction, and viscous heating are all coupled in a single gas evolution process. Therefore,
it is preferred to develop such a scheme with multi-dimensionality, and unified treatment of inviscid and dissipative terms. A high-order scheme does prefer a high-order
gas evolution model. Even with the rapid advances of high-order reconstruction techniques, the first-order dynamics of the Riemann solution becomes the bottleneck for
the further development of high-order schemes. In order to avoid the weakness of the
low order flux function, the development of high-order schemes relies heavily on the
weak solution of the original governing equations for the update of additional degree
of freedom, such as the non-conservative gradients of flow variables, which cannot be
physically valid in discontinuous regions. 相似文献
15.
High Order Finite Difference WENO Methods with Unequal-Sized Sub-Stencils for the Degasperis-Procesi Type Equations
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Jianfang Lin Yan Xu Huiwen Xue & Xinghui Zhong 《Communications In Computational Physics》2022,31(3):913-946
In this paper, we develop two finite difference weighted essentially
non-oscillatory (WENO) schemes with unequal-sized sub-stencils for solving the
Degasperis-Procesi (DP) and $\mu$-Degasperis-Procesi ($\mu$DP) equations, which contain
nonlinear high order derivatives, and possibly peakon solutions or shock waves. By
introducing auxiliary variable(s), we rewrite the DP equation as a hyperbolic-elliptic
system, and the $\mu$DP equation as a first order system. Then we choose a linear finite
difference scheme with suitable order of accuracy for the auxiliary variable(s), and
two finite difference WENO schemes with unequal-sized sub-stencils for the primal
variable. One WENO scheme uses one large stencil and several smaller stencils, and
the other WENO scheme is based on the multi-resolution framework which uses a series of unequal-sized hierarchical central stencils. Comparing with the classical WENO
scheme which uses several small stencils of the same size to make up a big stencil, both
WENO schemes with unequal-sized sub-stencils are simple in the choice of the stencil
and enjoy the freedom of arbitrary positive linear weights. Another advantage is that
the final reconstructed polynomial on the target cell is a polynomial of the same degree as the polynomial over the big stencil, while the classical finite difference WENO
reconstruction can only be obtained for specific points inside the target interval. Numerical tests are provided to demonstrate the high order accuracy and non-oscillatory
properties of the proposed schemes. 相似文献
16.
Arbitrarily High-Order (Weighted) Essentially Non-Oscillatory Finite Difference Schemes for Anelastic Flows on Staggered Meshes
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Siddhartha Mishra Carlos Paré s-Pulido & Kyle G. Pressel 《Communications In Computational Physics》2021,29(5):1299-1335
We propose a WENO finite difference scheme to approximate anelastic flows,
and scalars advected by them, on staggered grids. In contrast to existing WENO
schemes on staggered grids, the proposed scheme is designed to be arbitrarily high-order accurate as it judiciously combines ENO interpolations of velocities with WENO
reconstructions of spatial derivatives. A set of numerical experiments are presented
to demonstrate the increase in accuracy and robustness with the proposed scheme,
when compared to existing WENO schemes and state-of-the-art central finite difference schemes. 相似文献
17.
A High Order Sharp-Interface Method with Local Time Stepping for Compressible Multiphase Flows
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Angela Ferrari Claus-Dieter Munz & Bernhard Weigand 《Communications In Computational Physics》2011,9(1):205-230
In this paper, a new sharp-interface approach to simulate compressible
multiphase flows is proposed. The new scheme consists of a high order WENO finite volume scheme for solving the Euler equations coupled with a high order path-conservative
discontinuous Galerkin finite element scheme to evolve an indicator function
that tracks the material interface. At the interface our method applies ghost cells
to compute the numerical flux, as the ghost fluid method. However, unlike the original
ghost fluid scheme of Fedkiw et al. [15], the state of the ghost fluid is derived
from an approximate-state Riemann solver, similar to the approach proposed in [25],
but based on a much simpler formulation. Our formulation leads only to one single
scalar nonlinear algebraic equation that has to be solved at the interface, instead of
the system used in [25]. Away from the interface, we use the new general Osher-type
flux recently proposed by Dumbser and Toro [13], which is a simple but complete Riemann
solver, applicable to general hyperbolic conservation laws. The time integration
is performed using a fully-discrete one-step scheme, based on the approaches recently
proposed in [5, 7]. This allows us to evolve the system also with time-accurate local
time stepping. Due to the sub-cell resolution and the subsequent more restrictive
time-step constraint of the DG scheme, a local evolution for the indicator function is
applied, which is matched with the finite volume scheme for the solution of the Euler
equations that runs with a larger time step. The use of a locally optimal time step
avoids the introduction of excessive numerical diffusion in the finite volume scheme.
Two different fluids have been used, namely an ideal gas and a weakly compressible
fluid modeled by the Tait equation. Several tests have been computed to assess the
accuracy and the performance of the new high order scheme. A verification of our
algorithm has been carefully carried out using exact solutions as well as a comparison
with other numerical reference solutions. The material interface is resolved sharply
and accurately without spurious oscillations in the pressure field. 相似文献
18.
Guoliang Zhang & Tao Xiong 《Communications In Computational Physics》2022,32(1):126-155
We propose a high order finite difference linear scheme combined with a
high order bound preserving maximum-principle-preserving (MPP) flux limiter to
solve the incompressible flow system. For such problem with highly oscillatory structure but not strong shocks, our approach seems to be less dissipative and much less
costly than a WENO type scheme, and has high resolution due to a Hermite reconstruction. Spurious numerical oscillations can be controlled by the weak MPP flux
limiter. Numerical tests are performed for the Vlasov-Poisson system, the 2D guiding-center model and the incompressible Euler system. The comparison between the linear
and WENO type schemes, with and without the MPP flux limiter, will demonstrate the
good performance of our proposed approach. 相似文献
19.
Boyang Yu Hongtao Yang Yonghai Li & Guangwei Yuan 《Communications In Computational Physics》2022,31(5):1489-1524
We apply the monotonicity correction to the finite element method for the
anisotropic diffusion problems, including linear and quadratic finite elements on triangular meshes. When formulating the finite element schemes, we need to calculate
the integrals on every triangular element, whose results are the linear combination
of the two-point pairs. Then we decompose the integral results into the main and
remaining parts according to coefficient signs of two-point pairs. We apply the nonlinear correction to the positive remaining parts and move the negative remaining parts
to the right side of the finite element equations. Finally, the original stiffness matrix
can be transformed into a nonlinear M-matrix, and the corrected schemes have the
positivity-preserving property. We also give the monotonicity correction to the time
derivative term for the time-dependent problems. Numerical experiments show that
the corrected finite element method has monotonicity and maintains the convergence
order of the original schemes in $H^1$-norm and $L^2$-norm, respectively. 相似文献
20.
High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes
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Weijie Zhang Yulong Xing Yinhua Xia & Yan Xu 《Communications In Computational Physics》2022,31(3):771-815
In this paper, we propose a high-order accurate discontinuous Galerkin
(DG) method for the compressible Euler equations under gravitational fields on unstructured meshes. The scheme preserves a general hydrostatic equilibrium state and
provably guarantees the positivity of density and pressure at the same time. Comparing with the work on the well-balanced scheme for Euler equations with gravitation
on rectangular meshes, the extension to triangular meshes is conceptually plausible
but highly nontrivial. We first introduce a special way to recover the equilibrium state
and then design a group of novel variables at the interface of two adjacent cells, which
plays an important role in the well-balanced and positivity-preserving properties. One
main challenge is that the well-balanced schemes may not have the weak positivity
property. In order to achieve the well-balanced and positivity-preserving properties
simultaneously while maintaining high-order accuracy, we carefully design DG spatial discretization with well-balanced numerical fluxes and suitable source term approximation. For the ideal gas, we prove that the resulting well-balanced scheme, coupled with strong stability preserving time discretizations, satisfies a weak positivity
property. A simple existing limiter can be applied to enforce the positivity-preserving
property, without losing high-order accuracy and conservation. Extensive one- and
two-dimensional numerical examples demonstrate the desired properties of the proposed scheme, as well as its high resolution and robustness. 相似文献