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30
ADER schemes on adaptive triangular meshes for scalar conservation laws
 J. Comput. Phys
, 2005
"... Abstract. ADER schemes are recent finite volume methods for hyperbolic conservation laws, which can be viewed as generalizations of the classical first order Godunov method to arbitrary high orders. In the ADER approach, high order polynomial reconstruction from cell averages is combined with high o ..."
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Cited by 21 (6 self)
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Abstract. ADER schemes are recent finite volume methods for hyperbolic conservation laws, which can be viewed as generalizations of the classical first order Godunov method to arbitrary high orders. In the ADER approach, high order polynomial reconstruction from cell averages is combined with high order flux evaluation, where the latter is done by solving generalized Riemann problems across cell interfaces. Currently available nonlinear ADER schemes are restricted to Cartesian meshes. This paper proposes an adaptive nonlinear finite volume ADER method on unstructured triangular meshes for scalar conservation laws, which works with WENO reconstruction. To this end, a customized stencil selection scheme is developed, and the flux evaluation of previous ADER schemes is extended to triangular meshes. Moreover, an a posteriori error indicator is used to design the required adaption rules for the dynamic modification of the triangular mesh during the simulation. The expected convergence orders of the proposed ADER method are confirmed by numerical experiments for linear and nonlinear scalar conservation laws. Finally, the good performance of the adaptive ADER method, in particular its robustness and its enhanced flexibility, is further supported by numerical results concerning Burgers equation. 1
Derivative Riemann Solvers for Systems of Conservation
 Laws and ADER Methods. J. Comput Phys
"... In this paper we first briefly review the semianalytical method [20] for solving the Derivative Riemann Problem for systems of hyperbolic conservation laws with source terms. Next, we generalize it to hyperbolic systems for which the Riemann problem solution is not available. As an application exam ..."
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Cited by 14 (3 self)
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In this paper we first briefly review the semianalytical method [20] for solving the Derivative Riemann Problem for systems of hyperbolic conservation laws with source terms. Next, we generalize it to hyperbolic systems for which the Riemann problem solution is not available. As an application example we implement the new derivative Riemann solver in the highorder finitevolume ADER advection schemes. We provide numerical examples for the compressible Euler equations in two space dimensions which illustrate robustness and high accuracy of the resulting schemes. 1
High order conservative Lagrangian schemes with Lax–Wendroff type time discretization for the compressible Euler equations
 CODEN JCTPAH. ISSN 00219991 (print), 10902716 (electronic). URL http://www.sciencedirect.com/ science/article/pii/S0021999109004860. 543 Li:2004:FEM [LCW04] XiangGui
, 2009
"... In this paper, we explore the LaxWendro (LW) type time discretization as an alternative procedure to the high order RungeKutta time discretization adopted for the high order essentially nonoscillatory (ENO) Lagrangian schemes developed in [2, 4]. The LW time discretization is based on a Taylor ..."
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Cited by 13 (1 self)
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In this paper, we explore the LaxWendro (LW) type time discretization as an alternative procedure to the high order RungeKutta time discretization adopted for the high order essentially nonoscillatory (ENO) Lagrangian schemes developed in [2, 4]. The LW time discretization is based on a Taylor expansion in time, coupled with a local CauchyKowalewski procedure to utilize the partial dierential equation (PDE) repeatedly to convert all time derivatives to spatial derivatives, and then to discretize these spatial derivatives based on high order ENO reconstruction. Extensive numerical examples are presented, for both the second order spatial discretization using quadrilateral meshes [2] and third order spatial discretization using curvilinear meshes [4]. Comparing with the RungeKutta time discretization procedure, an advantage of the LW time discretization is the apparent saving in computational cost and memory requirement, at least for the two dimensional Euler equations that we have used in the numerical tests.
Very High Order PNPM Schemes on Unstructured Meshes for the Resistive Relativistic MHD Equations
, 903
"... In this paper we propose the first better than second order accurate method in space and time for the numerical solution of the resistive relativistic magnetohydrodynamics (RRMHD) equations on unstructured meshes in multiple space dimensions. The nonlinear system under consideration is purely hyperb ..."
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Cited by 13 (4 self)
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In this paper we propose the first better than second order accurate method in space and time for the numerical solution of the resistive relativistic magnetohydrodynamics (RRMHD) equations on unstructured meshes in multiple space dimensions. The nonlinear system under consideration is purely hyperbolic and contains a source term, the one for the evolution of the electric field, that becomes stiff for low values of the resistivity. For the spatial discretization we propose to use high order PNPM schemes as introduced in [10] for hyperbolic conservation laws and a high order accurate unsplit time discretization is achieved using the elementlocal spacetime discontinuous Galerkin approach proposed in [11] for onedimensional balance laws with stiff source terms. The divergence free character of the magnetic field is accounted for through the divergence cleaning procedure of Dedner et al. [7]. To validate our high order method we first solve some numerical test cases for which exact analytical reference solutions are known and we also show numerical convergence studies in the stiff limit of the RRMHD equations using PNPM schemes
FORCE Schemes on Unstructured Meshes II: Nonconservative Hyperbolic Systems
"... In this paper we propose a new high order accurate centered pathconservative method on unstructured triangular and tetrahedral meshes for the solution of multidimensional nonconservative hyperbolic systems, as they typically arise in the context of compressible multiphase flows. Our pathconserva ..."
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Cited by 7 (1 self)
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In this paper we propose a new high order accurate centered pathconservative method on unstructured triangular and tetrahedral meshes for the solution of multidimensional nonconservative hyperbolic systems, as they typically arise in the context of compressible multiphase flows. Our pathconservative centered scheme is an extension of the centered method recently proposed in [36] for conservation laws, to which it reduces if the system matrix is the Jacobian of a flux function. The main advantage in the proposed centered approach compared to upwind methods is that no information about the eigenstructure of the system or Roe averages are needed. The final fully discrete high order accurate formulation in space and time is obtained using the general framework of PN PM schemes proposed in [15], which unifies in one single general family of schemes classical finite volume and discontinuous Galerkin methods. We show applications of our high order centered method to the two and threedimensional BaerNunziato equations of compressible multiphase flows [3]. Key words: Nonconservative hyperbolic systems, centered schemes, unstructured meshes, high order finite volume and discontinuous Galerkin methods, compressible multiphase flow, Baer–Nunziato model
Arbitrary high order finite volume schemes for seismic . . .
, 2000
"... We present a new numerical method to solve the heterogeneous anelastic seismic wave equations with arbitrary high order of accuracy in space and time on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. Using the velocitystress formulation provides a l ..."
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Cited by 6 (1 self)
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We present a new numerical method to solve the heterogeneous anelastic seismic wave equations with arbitrary high order of accuracy in space and time on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. Using the velocitystress formulation provides a linear hyperbolic system of equations with source terms that is completed by additional equations for the anelastic functions including the strain history of the material. These additional equations result from the rheological model of the generalized Maxwell body and permit the incorporation of realistic attenuation properties of viscoelastic material accounting for the behaviour of elastic solids and viscous ßuids. The proposed method relies on the Finite Volume (FV) approach where cellaveraged quantities are evolved in time by computing numerical ßuxes at the element interfaces. The basic ingredient of the numerical ßux function is the solution of Generalized Riemann Problems at the element interfaces according to the ADER approach of Toro et al., where the initial data is piecewise polynomial instead of piecewise constant as it was in the original Þrst order FV scheme developed by Godunov. The ADER approach automatically produces a scheme of uniform high order of accuracy in space and time. The high order polynomials in space, needed as input for the numerical ßux function,
Improved Detection Criteria for the Multidimensional Optimal Order Detection (MOOD) on unstructured meshes with very highorder polynomials
, 2012
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ADERWENO Finite Volume Schemes with SpaceTime Adaptive Mesh Refinement
, 2014
"... We present the first high order onestep ADERWENO finite volume scheme with Adaptive Mesh Refinement (AMR) in multiple space dimensions. High order spatial accuracy is obtained through a WENO reconstruction, while a high order onestep time discretization is achieved using a local spacetime discon ..."
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Cited by 3 (0 self)
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We present the first high order onestep ADERWENO finite volume scheme with Adaptive Mesh Refinement (AMR) in multiple space dimensions. High order spatial accuracy is obtained through a WENO reconstruction, while a high order onestep time discretization is achieved using a local spacetime discontinuous Galerkin predictor method. Due to the onestep nature of the underlying scheme, the resulting algorithm is particularly well suited for an AMR strategy on spacetime adaptive meshes, i.e.with timeaccurate local time stepping. The AMR property has been implemented ’cellbycell’, with a standard treetype algorithm, while the scheme has been parallelized via the Message Passing Interface (MPI) paradigm. The new scheme has been tested over a wide range of examples for nonlinear systems of hyperbolic conservation laws, including the classical Euler equations of compressible gas dynamics and the equations of magnetohydrodynamics (MHD). High order in space and time have been confirmed via a numerical convergence study and a detailed analysis of the computational speedup with respect to highly refined uniform meshes is also presented. We also show test problems where the presented high order AMR scheme behaves clearly better than traditional second order AMR methods. The proposed scheme that combines for the first time high order ADER methods with space–time adaptive grids in two and three space dimensions is likely to become a useful tool in several fields of computational physics, applied mathematics and mechanics.
Lagrangian ADERWENO Finite Volume Schemes on Unstructured Triangular Meshes Based On Genuinely Multidimensional HLL Riemann Solvers
"... In this paper we use the genuinely multidimensional HLL Riemann solvers recently developed by Balsara et al. in [13] to construct a new class of computationally efficient high order Lagrangian ADERWENO onestep ALE finite volume schemes on unstructured triangular meshes. A nonlinear WENO reconstru ..."
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Cited by 2 (0 self)
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In this paper we use the genuinely multidimensional HLL Riemann solvers recently developed by Balsara et al. in [13] to construct a new class of computationally efficient high order Lagrangian ADERWENO onestep ALE finite volume schemes on unstructured triangular meshes. A nonlinear WENO reconstruction operator allows the algorithm to achieve high order of accuracy in space, while high order of accuracy in time is obtained by the use of an ADER timestepping technique based on a local spacetime Galerkin predictor. The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the grid, considering the entire Voronoi neighborhood of each node and allows for larger time steps than conventional onedimensional Riemann solvers. The results produced by the multidimensional Riemann solver are then used twice in our onestep ALE algorithm: first, as a node solver that assigns a unique velocity vector to each vertex, in order to preserve the continuity of the computational mesh; second, as a building block for genuinely multidimensional numerical flux evaluation that allows the scheme to run with larger time steps compared to conventional finite volume schemes that use classical onedimensional Riemann solvers in normal direction. The spacetime flux integral computation is carried out at the boundaries of each triangular spacetime control volume using the Simpson quadrature rule in space and GaussLegendre quadrature in time. A rezoning step may be necessary in order to overcome element overlapping or crossingover. Since our onestep ALE finite volume scheme is based directly on a spacetime conservation formulation of the governing PDE system, the remapping stage is not needed, making our algorithm a socalled direct ALE method. We apply the method presented in this article to two systems of hyperbolic conservation laws, namely the Euler equations of compressible gas dynamics and the