418 research outputs found
High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics
The paper develops high-order accurate physical-constraints-preserving finite
difference WENO schemes for special relativistic hydrodynamical (RHD)
equations, built on the local Lax-Friedrich splitting, the WENO reconstruction,
the physical-constraints-preserving flux limiter, and the high-order strong
stability preserving time discretization. They are extensions of the
positivity-preserving finite difference WENO schemes for the non-relativistic
Euler equations. However, developing physical-constraints-preserving methods
for the RHD system becomes much more difficult than the non-relativistic case
because of the strongly coupling between the RHD equations, no explicit
expressions of the primitive variables and the flux vectors, in terms of the
conservative vector, and one more physical constraint for the fluid velocity in
addition to the positivity of the rest-mass density and the pressure. The key
is to prove the convexity and other properties of the admissible state set and
discover a concave function with respect to the conservative vector replacing
the pressure which is an important ingredient to enforce the
positivity-preserving property for the non-relativistic case. Several one- and
two-dimensional numerical examples are used to demonstrate accuracy,
robustness, and effectiveness of the proposed physical-constraints-preserving
schemes in solving RHD problems with large Lorentz factor, or strong
discontinuities, or low rest-mass density or pressure etc.Comment: 39 pages, 13 figure
Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part I: The one-dimensional case
International audienceOne of the main issues in the field of numerical schemes is to ally robustness with accuracy. Considering gas dynamics, numerical approximations may generate negative density or pressure, which may lead to nonlinear instability and crash of the code. This phenomenon is even more critical using a Lagrangian formalism, the grid moving and being deformed during the calculation. Furthermore, most of the problems studied in this framework contain very intense rarefaction and shock waves. In this paper, the admissibility of numerical solutions obtained by high-order finite-volume-scheme-based methods, such as the discontinuous Galerkin (DG) method, the essentially non-oscillatory (ENO) and the weighted ENO (WENO) finite volume schemes, is addressed in the one-dimensional Lagrangian gas dynamics framework. After briefly recalling how to derive Lagrangian forms of the 1D gas dynamics system of equations, a discussion on positivity-preserving approximate Riemann solvers, ensuring first-order finite volume schemes to be positive, is then given. This study is conducted for both ideal gas and non ideal gas equations of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Mie-Grüneisen (MG) EOS, and relies on two different techniques: either a particular definition of the local approximation of the acoustic impedances arising from the approximate Riemann solver, or an additional time step constraint relative to the cell volume variation. Then, making use of the work presented in [89, 90, 22], this positivity study is extended to high-orders of accuracy, where new time step constraints are obtained, and proper limitation is required. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. This paper is the first part of a series of two. The whole analysis presented here is extended to the two-dimensional case in [85], and proves to fit a wide range of numerical schemes in the literature, such as those presented in [19, 64, 15, 82, 84]
Lagrangian ADER-WENO 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. to construct a new class of
computationally efficient high order Lagrangian ADER-WENO one-step 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 time-stepping technique based on a local space-time 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 one-dimensional Riemann solvers. The
results produced by the multidimensional Riemann solver are then used twice in
our one-step 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
one-dimensional Riemann solvers in normal direction. A rezoning step may be
necessary in order to overcome element overlapping or crossing-over. 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 equations
of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to
fourth order of accuracy in space and time have been carried out. Several
numerical test problems have been solved to validate the new approach
Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders
One of the main issues in the field of numerical schemes is to ally robustness with accuracy. Considering gas dynamics, numerical approximations may generate negative density or pressure, which may lead to nonlinear instability and crash of the code. This phenomenon is even more critical using a Lagrangian formalism, the grid moving and being deformed during the calculation. Furthermore, most of the problems studied in this framework contain very intense rarefaction and shock waves. In this paper, the admissibility of numerical solutions obtained by high-order finite-volume-scheme-based methods, such as the discontinuous Galerkin (DG) method, the essentially non-oscillatory (ENO) and the weighted ENO (WENO) finite volume schemes, is addressed in this Lagrangian gas dynamics framework. To this end, we first focus on the one-dimensional case. After briefly recalling how to derive Lagrangian forms of the gas dynamics system of equations, a discussion on positivity-preserving approximate Riemann solvers, ensuring first-order finite volume schemes to be positive, is then given. This study is conducted for both ideal gas and non ideal gas equations of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Mie-Grüneisen (MG) EOS. It enables us to derive time step conditions ensuring the desired positivity property, as well as L 1 stability of the specific volume and total energy over the domain. Then, making use of the work presented in [74, 75, 15], this positivity study is extended to high-orders of accuracy, where new time step constraints are obtained, and proper limitation is required. This whole analysis is finally applied to the two-dimensional case, and shown to fit a wide range of numerical schemes in the literature, such as the GLACE scheme [12], the EUCCLHYD scheme [55], the GLACE scheme on conical meshes [8], and the LCCDG method [72]. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. Finally, let us emphasize that even if this paper is concerned with purely Lagrangian schemes, the theory developed is of fundamental importance for any methods relying on a purely Lagrangian step, as ALE methods or non-direct Euler schemes
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