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198
A subgridscale deconvolution approach for shock capturing
 J.C.P
"... We develop a method for the modeling of flow discontinuities which can arise as weak solutions of inviscid conservation laws. Due to its similarity with recently proposed approximate deconvolution models for largeeddy simulation, the method potentially allows for a unified treatment of flow discont ..."
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Cited by 14 (0 self)
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We develop a method for the modeling of flow discontinuities which can arise as weak solutions of inviscid conservation laws. Due to its similarity with recently proposed approximate deconvolution models for largeeddy simulation, the method potentially allows for a unified treatment of flow discontinuities and turbulent subgrid scales. A filtering approach is employed since for the filtered evolution equations the solution is smooth and can be solved for by standard central finitedifference schemes without special consideration of discontinuities. A sufficiently accurate representation of the filtered nonlinear combination of discontinuous solution components which arise from the convection term can be obtained by a regularized deconvolution applied to the filtered solution. For stable integration the evolution equations are supplemented by a relaxation regularization based on a secondary filter operation and a relaxation parameter. An estimate for the relaxation parameter is provided. The method is related to the spectral vanishingviscosity method and the regularized Chapman–Enskog expansion method for conservation laws. We detail the approach and demonstrate its efficiency with the inviscid and viscous Burgers equations, the isothermal shock problem, and the onedimensional Euler
Numerical Hydrodynamics In Special Relativity
, 1999
"... This review is concerned with a discussion of numerical methods for the solution of the equations of special relativistic hydrodynamics (SRHD). Particular emphasis is put on a comprehensive review of the application of highresolution shockcapturing methods in SRHD. Results obtained with different ..."
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Cited by 13 (0 self)
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This review is concerned with a discussion of numerical methods for the solution of the equations of special relativistic hydrodynamics (SRHD). Particular emphasis is put on a comprehensive review of the application of highresolution shockcapturing methods in SRHD. Results obtained with different numerical SRHD methods are compared, and two astrophysical applications of SRHD flows are discussed. An evaluation of the various numerical methods is given and future developments are analyzed.
Arbitrary high order discontinuous Galerkin schemes. Numerical Methods for Hyperbolic and Kinetic Problems
 Goudon & E. Sonnendrucker eds). IRMA Series in Mathematics and Theoretical Physics
, 2005
"... Abstract. In this paper we apply the ADER one step time discretization to the Discontinuous Galerkin framework for hyperbolic conservation laws. In the case of linear hyperbolic systems we obtain a quadraturefree explicit singlestep scheme of arbitrary order of accuracy in space and time on Cartes ..."
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Cited by 13 (1 self)
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Abstract. In this paper we apply the ADER one step time discretization to the Discontinuous Galerkin framework for hyperbolic conservation laws. In the case of linear hyperbolic systems we obtain a quadraturefree explicit singlestep scheme of arbitrary order of accuracy in space and time on Cartesian and triangular meshes. The ADERDG scheme does not need more memory than a first order explicit Euler timestepping scheme. This becomes possible because of an extensive use of the governing equations inside the numerical scheme. In the nonlinear case, quadrature of the ADERDG scheme in space and time is performed with Gaussian quadrature formulae of suitable order of accuracy. We show numerical convergence results for the linearized Euler equations up to 10th order of accuracy in space and time on Cartesian and triangular meshes. Numerical results for the nonlinear Euler equations up to 6th order of accuracy in space and time are provided as well. In this paper we also show the possibility of applying a linear reconstruction operator of the order 3N + 2 to the degrees of freedom of the DG method resulting in a numerical scheme of the order 3N + 3 on Cartesian grids where N is the order of the original basis functions before reconstruction. Key words. Discontinuous Galerkin finite elements, ADER approach AMS. 65M60 1.
Axisymmetric Numerical Relativity
, 2005
"... Chapters 2, 3 and 6 contain work done in collaboration with my supervisor and published in a joint paper [119]. The dynamical shift conditions in chapter 6 are a later addition by myself. The remaining chapters are my own work. All computer programmes were written by myself unless otherwise stated. ..."
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Cited by 12 (6 self)
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Chapters 2, 3 and 6 contain work done in collaboration with my supervisor and published in a joint paper [119]. The dynamical shift conditions in chapter 6 are a later addition by myself. The remaining chapters are my own work. All computer programmes were written by myself unless otherwise stated. c○Oliver Rinne, 2005 This thesis is concerned with formulations of the Einstein equations in axisymmetric spacetimes which are suitable for numerical evolutions. The common basis for our formulations is provided by the (2+1)+1 formalism. General matter sources and rotational degrees of freedom are included. A first evolution system adopts elliptic gauge conditions arising from maximal slicing and conformal flatness. The numerical implementation is based on the finitedifference approach, using a Multigrid algorithm for the elliptic equations and the method of lines for the hyperbolic evolution equations.
Shock capturing and front tracking methods for granular avalanches
 J. Comput. Phys
, 2002
"... Shock formations are observed in granular avalanches when supercritical flow merges into a region of subcritical flow. In this paper we employ a shockcapturing numerical scheme for the onedimensional Savage–Hutter theory of granular flow to describe this phenomenon. A Lagrangian moving mesh scheme ..."
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Cited by 10 (4 self)
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Shock formations are observed in granular avalanches when supercritical flow merges into a region of subcritical flow. In this paper we employ a shockcapturing numerical scheme for the onedimensional Savage–Hutter theory of granular flow to describe this phenomenon. A Lagrangian moving mesh scheme applied to the nonconservative form of the equations reproduces smooth solutions of these free boundary problems very well, but fails when shocks are formed. A nonoscillatory central (NOC) difference scheme with TVD limiter or WENO cell reconstruction for the conservative equations is therefore introduced. For the avalanche free boundary problems it must be combined with a fronttracking method, developed here, to properly describe the margin evolution. It is found that this NOC scheme combined with the fronttracking module reproduces both the shock wave and the smooth solution accurately. A piecewise quadratic WENO reconstruction improves the smoothness of the solution near local extrema. The schemes are checked against exact solutions for (1) an upward moving shock wave, (2) the motion of a parabolic cap down an inclined plane, and (3) the motion of a parabolic cap down a curved slope ending in a flat runout region, where a shock is formed as the avalanche comes to a halt. c ○ 2002 Elsevier Science Key Words: granular avalanche; shockcapturing; nonoscillatory central scheme; free moving boundary; fronttracking. 1.
Numerical Methods for The Simulation of The Settling of Flocculated Suspensions
"... For one space dimension, the phenomenological theory of sedimentation of flocculated suspensions yields a model that consists of an initialboundary value problem for a second order partial differential equation of mixed hyperbolicparabolic type. Due to the mixed hyperbolicparabolic nature of the ..."
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Cited by 9 (6 self)
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For one space dimension, the phenomenological theory of sedimentation of flocculated suspensions yields a model that consists of an initialboundary value problem for a second order partial differential equation of mixed hyperbolicparabolic type. Due to the mixed hyperbolicparabolic nature of the model, its solutions may be discontinuous and difficulties arise if one tries to construct these solutions by classical numerical methods. In this paper we present and elaborate on numerical methods that can be used to correctly simulate this model, i.e., conservative methods satisfying a discrete entropy principle. Included in our discussion are finite difference methods and methods based on operator splitting. In particular, the operator splitting methods are used to simulate the settling of flocculated suspensions.
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
Conservative semiLagrangian advection on adaptive unstructured meshes
 Num. Meth. Partial Diff. Eq
, 2003
"... Abstract. A conservative semiLagrangian method is designed in order to solve linear advection equations in two space variables. The advection scheme works with finite volumes on an unstructured mesh, which is given by a Voronoi diagram. Moreover, the mesh is subject to adaptive modifications during ..."
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Cited by 8 (2 self)
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Abstract. A conservative semiLagrangian method is designed in order to solve linear advection equations in two space variables. The advection scheme works with finite volumes on an unstructured mesh, which is given by a Voronoi diagram. Moreover, the mesh is subject to adaptive modifications during the simulation, which serves to effectively combine good approximation quality with small computational costs. The required adaption rules for the refinement and the coarsening of the mesh rely on a customized error indicator. The implementation of boundary conditions is addressed. Numerical results finally confirm the good performance of the proposed conservative and adaptive advection scheme. 1
A lownumerical dissipation patchbased adaptive mesh refinement method for largeeddy simulation of compressible flows
, 2005
"... This paper presents a hybrid finitedi#erence/weighted essentially nonoscillatory (WENO) method for largeeddy simulation of compressible flows with lownumerical dissipation schemes and structured adaptive mesh refinement (SAMR). A conservative fluxbased approach is described, encompassing the ..."
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Cited by 8 (4 self)
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This paper presents a hybrid finitedi#erence/weighted essentially nonoscillatory (WENO) method for largeeddy simulation of compressible flows with lownumerical dissipation schemes and structured adaptive mesh refinement (SAMR). A conservative fluxbased approach is described, encompassing the cases of scheme alternation and internal mesh interfaces resulting from SAMR. An explicit centered scheme is used in turbulent flow regions while a WENO scheme is employed to capture shocks.
A MULTIWAVE APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD BASED ON RELAXATION II NUMERICAL IMPLEMENTATION WITH 3 AND 5 WAVES
"... Abstract. In the first part of this work ([5]), we introduced an approximate Riemann solver for onedimensional ideal MHD derived from a relaxation system. We gave sufficient conditions for the solver to satisfy discrete entropy inequalities, and to preserve positivity of density and internal energy ..."
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Cited by 8 (4 self)
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Abstract. In the first part of this work ([5]), we introduced an approximate Riemann solver for onedimensional ideal MHD derived from a relaxation system. We gave sufficient conditions for the solver to satisfy discrete entropy inequalities, and to preserve positivity of density and internal energy. In this paper we consider the practical implementation, and derive explicit wave speed estimates satisfying the stability conditions of [5]. We present a 3wave solver that well resolves fast waves and material contacts, and a 5wave solver that accurately resolves the cases when two eigenvalues coincide. A full 7wave solver, which is highly accurate on all types of waves, will be described in a followup paper. We test the solvers on onedimensional shock tube data and smooth shear waves. (1.1) (1.2)