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11
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 9 (4 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 6 (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
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 2 (0 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
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 1 (0 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
A Unified Framework for the Construction of OneStep FiniteVolume and Discontinuous Galerkin Schemes on Unstructured Meshes
"... In this article a conservative leastsquares polynomial reconstruction operator is applied to the discontinuous Galerkin method. In a first instance, piecewise polynomials of degree N are used as test functions as well as to represent the data in each element at the beginning of a time step. The tim ..."
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In this article a conservative leastsquares polynomial reconstruction operator is applied to the discontinuous Galerkin method. In a first instance, piecewise polynomials of degree N are used as test functions as well as to represent the data in each element at the beginning of a time step. The time evolution of these data and the flux computation, however, are then done with a different set of piecewise polynomials of degree M ≥ N, which are reconstructed from the underlying polynomials of degree N. This approach yields a general, unified framework that contains as two special cases classical high order finite volume (FV) schemes (N = 0) as well as the usual discontinuous Galerkin (DG) method (N = M). In the first case, the polynomial is reconstructed from cell averages, for the latter, the reconstruction reduces to the identity operator. A completely new class of numerical schemes is generated by choosing N ̸ = 0 and M> N. The reconstruction operator is implemented for arbitrary polynomial degrees N and M on unstructured triangular and tetrahedral meshes in two and three space dimensions.
Explicit Finite Volume Schemes of Arbitrary High Order of Accuracy for Hyperbolic Systems with Stiff Source Terms
, 2007
"... In this article we propose a new class of finite volume schemes of arbitrary accuracy in space and time for systems of hyperbolic balance laws with stiff source terms. The new class of schemes is based on a three stage procedure. First, in order to achieve high order accuracy in space, a nonlinear w ..."
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In this article we propose a new class of finite volume schemes of arbitrary accuracy in space and time for systems of hyperbolic balance laws with stiff source terms. The new class of schemes is based on a three stage procedure. First, in order to achieve high order accuracy in space, a nonlinear weighted essentially nonoscillatory reconstruction procedure is applied to the cell averages at the current time level. Second, the temporal evolution of the resulting reconstruction polynomials is computed locally inside each cell exploiting directly the full system of governing equations. In previous ADER schemes, this was achieved via the CauchyKovalewski procedure, where the governing equation is repeatedly differentiated with respect to space and time to construct a Taylor series expansion of the local solution. As the CauchyKovalewski procedure is based on Taylor series expansions, it is not able to handle systems with stiff source terms since the Taylor series diverges for this case. Therefore, in this article, we present a new strategy that replaces the CauchyKovalewski procedure for high order time interpolation: we present a special local spacetime
MAGNÉTIQUE AUTOGÉNÉRÉ
, 2011
"... Ce travail est consacré à la construction de méthodes numériques permettant la simulation de processus d’implosion de coquilles en fusion par confinement inertiel (FCI) avec prise en compte des termes de champ magnétique autogénéré. Dans ce document, on commence par décrire le modèle de magnétohydr ..."
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Ce travail est consacré à la construction de méthodes numériques permettant la simulation de processus d’implosion de coquilles en fusion par confinement inertiel (FCI) avec prise en compte des termes de champ magnétique autogénéré. Dans ce document, on commence par décrire le modèle de magnétohydrodynamique résistive à deux températures considéré ainsi que les relations de fermeture utilisées. Le système d’équations ainsi obtenu est alors divisé en soussystèmes selon la nature de l’opérateur mathématique sousjacent pour lesquels l’on propose ensuite des schémas numériques adaptés. On insiste notamment sur le développement de schémas volumes finis pour l’opérateur hyperbolique, ce dernier correspondant aux équations d’Euler ou de la magnétohydrodynamique idéale selon que l’on tienne compte ou non des termes de champ magnétique. Plus précisement, on propose une nouvelle classe de schémas d’ordre élevé à directions alternées construits dans le formalisme Lagrange + projection sur grille cartésienne qui présentent l’originalité d’être particulièrement bien adaptés aux calculateurs modernes grâce, entre autres,
ON SOURCE TERMS AND BOUNDARY CONDITIONS USING ARBITRARY HIGH ORDER DISCONTINUOUS GALERKIN SCHEMES
"... This article is devoted to the discretization of source terms and boundary conditions using discontinuous Galerkin schemes with an arbitrary high order of accuracy in space and time for the solution of hyperbolic conservation laws on unstructured triangular meshes. The building block of the method i ..."
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This article is devoted to the discretization of source terms and boundary conditions using discontinuous Galerkin schemes with an arbitrary high order of accuracy in space and time for the solution of hyperbolic conservation laws on unstructured triangular meshes. The building block of the method is a particular numerical flux function at the element interfaces based on the solution of Generalized Riemann Problems (GRPs) with piecewise polynomial initial data. The solution of the generalized Riemann problem, originally introduced by Toro and Titarev in a finite volume context, provides simultaneously a numerical flux function as well as a time integration method. The resulting scheme is extremely local since it integrates the PDE from one time step to the successive one in a single step using only information from the direct side neighbors. Since source terms are directly incorporated into the numerical flux via the solution of the GRP, our very high order accurate method is also able to maintain very well smooth steadystate solutions of PDEs with source terms, similar to the socalled wellbalanced schemes which are usually specially designed for this purpose. Boundary conditions are imposed solving inverse generalized Riemann problems. Furthermore, we show numerical evidence proving that by using very high order schemes together with high order polynomial representations of curved boundaries, high quality solutions can be obtained on very coarse meshes.