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1.
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
2.
Ruo Li & Wei Zhong 《Communications In Computational Physics》2022,31(2):548-592
Existing mapped WENO schemes can hardly prevent spurious oscillations
while preserving high resolutions at long output times. We reveal in this paper the essential reason of such phenomena. It is actually caused by that the mapping function
in these schemes can not preserve the order of the nonlinear weights of the stencils.
The nonlinear weights may be increased for non-smooth stencils and be decreased for
smooth stencils. It is then indicated to require the set of mapping functions to be order-preserving in mapped WENO schemes. Therefore, we propose a new mapped WENO
scheme with a set of mapping functions to be order-preserving which exhibits a remarkable advantage over the mapped WENO schemes in references. For long output
time simulations of the one-dimensional linear advection equation, the new scheme
has the capacity to attain high resolutions and avoid spurious oscillations near discontinuities meanwhile. In addition, for the two-dimensional Euler problems with strong
shock waves, the new scheme can significantly reduce the numerical oscillations. 相似文献
3.
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
4.
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
5.
Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes
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In this article we present a new class of high order accurate ArbitraryEulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes. A WENO reconstruction algorithm is used to achieve high
order accuracy in space and a high order one-step time discretization is achieved by
using the local space-time Galerkin predictor proposed in [25]. For that purpose, a
new element-local weak formulation of the governing PDE is adopted on moving
space-time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes.
Moreover, a polynomial mapping defined by the same local space-time basis functions
as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the
final ALE one-step finite volume scheme uses moving triangular meshes with straight
edges. This is possible in the ALE framework, which allows a local mesh velocity that
is different from the local fluid velocity. We present numerical convergence rates for
the schemes presented in this paper up to sixth order of accuracy in space and time and
show some classical numerical test problems for the two-dimensional Euler equations
of compressible gas dynamics. 相似文献
6.
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
7.
High Order Schemes on Three-Dimensional General Polyhedral Meshes — Application to the Level Set Method
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In this article, we detail the methodology developed to construct arbitrarily high order schemes — linear and WENO — on 3D mixed-element unstructured
meshes made up of general convex polyhedral elements. The approach is tailored
specifically for the solution of scalar level set equations for application to incompressible two-phase flow problems. The construction of WENO schemes on 3D unstructured meshes is notoriously difficult, as it involves a much higher level of complexity
than 2D approaches. This due to the multiplicity of geometrical considerations introduced by the extra dimension, especially on mixed-element meshes. Therefore, we
have specifically developed a number of algorithms to handle mixed-element meshes
composed of convex polyhedra with convex polygonal faces. The contribution of this
work concerns several areas of interest: the formulation of an improved methodology
in 3D, the minimisation of computational runtime in the implementation through the
maximum use of pre-processing operations, the generation of novel methods to handle complex 3D mixed-element meshes and finally the application of the method to
the transport of a scalar level set. 相似文献
8.
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
9.
Adaptive Order WENO Reconstructions for the Semi-Lagrangian Finite Difference Scheme for Advection Problem
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Jiajie Chen Xiaofeng Cai Jianxian Qiu & Jing-Mei Qiu 《Communications In Computational Physics》2021,30(1):67-96
We present a new conservative semi-Lagrangian finite difference weighted
essentially non-oscillatory scheme with adaptive order. This is an extension of the
conservative semi-Lagrangian (SL) finite difference WENO scheme in [Qiu and Shu,
JCP, 230 (4) (2011), pp. 863-889], in which linear weights in SL WENO framework
were shown not to exist for variable coefficient problems. Hence, the order of accuracy is not optimal from reconstruction stencils. In this paper, we incorporate a recent
WENO adaptive order (AO) technique [Balsara et al., JCP, 326 (2016), pp. 780-804]
to the SL WENO framework. The new scheme can achieve an optimal high order of
accuracy, while maintaining the properties of mass conservation and non-oscillatory
capture of solutions from the original SL WENO. The positivity-preserving limiter is
further applied to ensure the positivity of solutions. Finally, the scheme is applied to
high dimensional problems by a fourth-order dimensional splitting. We demonstrate
the effectiveness of the new scheme by extensive numerical tests on linear advection
equations, the Vlasov-Poisson system, the guiding center Vlasov model as well as the
incompressible Euler equations. 相似文献
10.
Comparative Study of Three High Order Schemes for LES of Temporally Evolving Mixing Layers
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Helen C. Yee Bjorn Sjö green & Abdellah Hadjadj 《Communications In Computational Physics》2012,12(5):1603-1622
Three high order shock-capturing schemes are compared for large eddy
simulations (LES) of temporally evolving mixing layers for different convective Mach
numbers ranging from the quasi-incompressible regime to highly compressible supersonic regime. The considered high order schemes are fifth-order WENO (WENO5),
seventh-order WENO (WENO7) and the associated eighth-order central spatial base
scheme with the dissipative portion of WENO7 as a nonlinear post-processing filter
step (WENO7fi). This high order nonlinear filter method of Yee & Sjogreen is designed for accurate and efficient simulations of shock-free compressible turbulence,
turbulence with shocklets and turbulence with strong shocks with minimum tuning
of scheme parameters. The LES results by WENO7fi using the same scheme parameter agree well with experimental results compiled by Barone et al., and published
direct numerical simulations (DNS) work of Rogers & Moser and Pantano & Sarkar,
whereas results by WENO5 and WENO7 compare poorly with experimental data and
DNS computations. 相似文献
11.
Hong Luo Luqing Luo & Robert Nourgaliev 《Communications In Computational Physics》2012,12(5):1495-1519
A reconstruction-based discontinuous Galerkin (RDG(P1P2)) method, avariant of P1P2 method, is presented for the solution of the compressible Euler equations on arbitrary grids. In this method, an in-cell reconstruction, designed to enhance the accuracy of the discontinuous Galerkin method, is used to obtain a quadratic polynomial solution (P2) from the underlying linear polynomial (P1) discontinuous Galerkin solution using a least-squares method. The stencils used in the reconstruction involve only the von Neumann neighborhood (face-neighboring cells) and are compact and consistent with the underlying DG method. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG(P1P2) method is third-order accurate, and outperforms the third-order DG method (DG(P2)) in terms of both computing costs and storage requirements. 相似文献
12.
Yifei Wan & Yinhua Xia 《Communications In Computational Physics》2023,33(5):1270-1331
For steady Euler equations in complex boundary domains, high-order shockcapturing schemes usually suffer not only from the difficulty of steady-state convergence but also from the problem of dealing with physical boundaries on Cartesian
grids to achieve uniform high-order accuracy. In this paper, we utilize a fifth-order
finite difference hybrid WENO scheme to simulate steady Euler equations, and the
same fifth-order WENO extrapolation methods are developed to handle the curved
boundary. The values of the ghost points outside the physical boundary can be obtained by applying WENO extrapolation near the boundary, involving normal derivatives acquired by the simplified inverse Lax-Wendroff procedure. Both equivalent expressions involving curvature and numerical differentiation are utilized to transform
the tangential derivatives along the curved solid wall boundary. This hybrid WENO
scheme is robust for steady-state convergence and maintains high-order accuracy in
the smooth region even with the solid wall boundary condition. Besides, the essentially non-oscillation property is achieved. The numerical spectral analysis also shows
that this hybrid WENO scheme has low dispersion and dissipation errors. Numerical
examples are presented to validate the high-order accuracy and robust performance of
the hybrid scheme for steady Euler equations in curved domains with Cartesian grids. 相似文献
13.
In this paper, we introduce a new type of troubled-cell indicator to improve
hybrid weighted essentially non-oscillatory (WENO) schemes for solving the hyperbolic conservation laws. The hybrid WENO schemes selectively adopt the high-order
linear upwind scheme or the WENO scheme to avoid the local characteristic decompositions and calculations of the nonlinear weights in smooth regions. Therefore,
they can reduce computational cost while maintaining non-oscillatory properties in
non-smooth regions. Reliable troubled-cell indicators are essential for efficient hybrid
WENO methods. Most of troubled-cell indicators require proper parameters to detect
discontinuities precisely, but it is very difficult to determine the parameters automatically. We develop a new troubled-cell indicator derived from the mean value theorem
that does not require any variable parameters. Additionally, we investigate the characteristics of indicator variable; one of the conserved properties or the entropy is considered as indicator variable. Detailed numerical tests for 1D and 2D Euler equations are
conducted to demonstrate the performance of the proposed indicator. The results with
the proposed troubled-cell indicator are in good agreement with pure WENO schemes.
Also the new indicator has advantages in the computational cost compared with the
other indicators. 相似文献
14.
A Comparative Study of Rosenbrock-Type and Implicit Runge-Kutta Time Integration for Discontinuous Galerkin Method for Unsteady 3D Compressible Navier-Stokes equations
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Xiaodong Liu Yidong Xia Hong Luo & Lijun Xuan 《Communications In Computational Physics》2016,20(4):1016-1044
A comparative study of two classes of third-order implicit time integration
schemes is presented for a third-order hierarchical WENO reconstructed discontinuous
Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes
equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3)
scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential
algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme,
a remarkable feature of the ROW schemes is that, they only require one approximate
Jacobian matrix calculation every time step, thus considerably reducing the overall
computational cost. A variety of test cases, ranging from inviscid flows to DNS of
turbulent flows, are presented to assess the performance of these schemes. Numerical
experiments demonstrate that the third-order ROW scheme for the DAEs of index-2
can not only achieve the designed formal order of temporal convergence accuracy in
a benchmark test, but also require significantly less computing time than its ESDIRK3
counterpart to converge to the same level of discretization errors in all of the flow
simulations in this study, indicating that the ROW methods provide an attractive alternative
for the higher-order time-accurate integration of the unsteady compressible
Navier-Stokes equations. 相似文献
15.
High-Order Gas-Kinetic Scheme in Curvilinear Coordinates for the Euler and Navier-Stokes Solutions
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Liang Pan & Kun Xu 《Communications In Computational Physics》2020,28(4):1321-1351
The high-order gas-kinetic scheme (HGKS) has achieved success in simulating compressible flows with Cartesian meshes. To study the flow problems in general
geometries, such as the flow over a wing-body, the development of HGKS in general
curvilinear coordinates becomes necessary. In this paper, a two-stage fourth-order gas-kinetic scheme is developed for the Euler and Navier-Stokes solutions in the curvilinear coordinates from one-dimensional to three-dimensional computations. Based on
the coordinate transformation, the kinetic equation is transformed first to the computational space, and the flux function in the gas-kinetic scheme is obtained there and
is transformed back to the physical domain for the update of flow variables inside
each control volume. To achieve the expected order of accuracy, the dimension-by-dimension reconstruction based on the WENO scheme is adopted in the computational domain, where the reconstructed variables are the cell averaged Jacobian and the
Jacobian-weighted conservative variables. In the two-stage fourth-order gas-kinetic
scheme, the point values as well as the spatial derivatives of conservative variables at
Gaussian quadrature points have to be used in the evaluation of the time dependent
flux function. The point-wise conservative variables are obtained by ratio of the above
reconstructed data, and the spatial derivatives are reconstructed through orthogonalization in physical space and chain rule. A variety of numerical examples from the
accuracy tests to the solutions with strong discontinuities are presented to validate the
accuracy and robustness of the current scheme for both inviscid and viscous flows.
The precise satisfaction of the geometrical conservation law in non-orthogonal mesh is
also demonstrated through the numerical example. 相似文献
16.
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
17.
Weighted essentially non-oscillatory (WENO) methods have been developed to simultaneously provide robust shock-capturing in compressible fluid flow and
avoid excessive damping of fine-scale flow features such as turbulence. Under certain conditions in compressible turbulence, however, numerical dissipation remains
unacceptably high even after optimization of the linear component that dominates
in smooth regions. Of the nonlinear error that remains, we demonstrate that a large
fraction is generated by a "synchronization deficiency" that interferes with the expression of theoretically predicted numerical performance characteristics when the WENO
adaptation mechanism is engaged. This deficiency is illustrated numerically in simulations of a linearly advected sinusoidal wave and the Shu-Osher problem [J. Comput. Phys., 83 (1989), pp. 32-78]. It is shown that attempting to correct this deficiency
through forcible synchronization results in violation of conservation. We conclude
that, for the given choice of candidate stencils, the synchronization deficiency cannot
be adequately resolved under the current WENO smoothness measurement technique. 相似文献
18.
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
19.
Fourth Order Difference Approximations for Space Riemann-Liouville Derivatives Based on Weighted and Shifted Lubich Difference Operators
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High order discretization schemes play more important role in fractional operators
than classical ones. This is because usually for classical derivatives the stencil
for high order discretization schemes is wider than low order ones; but for fractional
operators the stencils for high order schemes and low order ones are the same. Then
using high order schemes to solve fractional equations leads to almost the same computational
cost as first order schemes but the accuracy is greatly improved. Using
the fractional linear multistep methods, Lubich obtains the ν-th order (ν≤6) approximations
of the α-th derivative (α>0) or integral (α<0) [Lubich, SIAM J. Math. Anal.,
17, 704-719, 1986], because of the stability issue the obtained scheme can not be directly
applied to the space fractional operator with α∈(1,2) for time dependent problem. By
weighting and shifting Lubich's 2nd order discretization scheme, in [Chen & Deng,
SINUM, arXiv:1304.7425] we derive a series of effective high order discretizations for
space fractional derivative, called WSLD operators there. As the sequel of the previous
work, we further provide new high order schemes for space fractional derivatives
by weighting and shifting Lubich's 3rd and 4th order discretizations. In particular,
we prove that the obtained 4th order approximations are effective for space fractional
derivatives. And the corresponding schemes are used to solve the space fractional
diffusion equation with variable coefficients. 相似文献
20.
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