81 research outputs found
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
Multidimensional HLLE Riemann solver; Application to Euler and Magnetohydrodynamic Flows
In this work we present a general strategy for constructing multidimensional
Riemann solvers with a single intermediate state, with particular attention
paid to detailing the two-dimensional Riemann solver. This is accomplished by
introducing a constant resolved state between the states being considered,
which introduces sufficient dissipation for systems of conservation laws.
Closed form expressions for the resolved fluxes are also provided to facilitate
numerical implementation. The Riemann solver is proved to be positively
conservative for the density variable; the positivity of the pressure variable
has been demonstrated for Euler flows when the divergence in the fluid
velocities is suitably restricted so as to prevent the formation of cavitation
in the flow.
We also focus on the construction of multidimensionally upwinded electric
fields for divergence-free magnetohydrodynamical flows. A robust and efficient
second order accurate numerical scheme for two and three dimensional Euler and
magnetohydrodynamic flows is presented. The scheme is built on the current
multidimensional Riemann solver. The number of zones updated per second by this
scheme on a modern processor is shown to be cost competitive with schemes that
are based on a one-dimensional Riemann solver. However, the present scheme
permits larger timesteps
A Two-dimensional HLLC Riemann Solver for Conservation Laws : Application to Euler and MHD Flows
In this paper we present a genuinely two-dimensional HLLC Riemann solver. On
logically rectangular meshes, it accepts four input states that come together
at an edge and outputs the multi-dimensionally upwinded fluxes in both
directions. This work builds on, and improves, our prior work on
two-dimensional HLL Riemann solvers. The HLL Riemann solver presented here
achieves its stabilization by introducing a constant state in the region of
strong interaction, where four one-dimensional Riemann problems interact
vigorously with one another. A robust version of the HLL Riemann solver is
presented here along with a strategy for introducing sub-structure in the
strongly-interacting state. Introducing sub-structure turns the two-dimensional
HLL Riemann solver into a two-dimensional HLLC Riemann solver. The
sub-structure that we introduce represents a contact discontinuity which can be
oriented in any direction relative to the mesh.
The Riemann solver presented here is general and can work with any system of
conservation laws. We also present a second order accurate Godunov scheme that
works in three dimensions and is entirely based on the present multidimensional
HLLC Riemann solver technology. The methods presented are cost-competitive with
traditional higher order Godunov schemes
Multidimensional Riemann Problem with Self-Similar Internal Structure – Part III– A Multidimensional Analogue of the HLLI Riemann Solver for Conservative Hyperbolic Systems
Just as the quality of a one-dimensional approximate Riemann solver is improved by the inclusion of internal sub-structure, the quality of a multidimensional Riemann solver is also similarly improved. Such multidimensional Riemann problems arise when multiple states come together at the vertex of a mesh. The interaction of the resulting one-dimensional Riemann problems gives rise to a strongly-interacting state. We wish to endow this strongly-interacting state with physically-motivated sub-structure. The fastest way of endowing such sub-structure consists of making a multidimensional extension of the HLLI Riemann solver for hyperbolic conservation laws. Presenting such a multidimensional analogue of the HLLI Riemann solver with linear sub-structure for use on structured meshes is the goal of this work. The multidimensional MuSIC Riemann solver documented here is universal in the sense that it can be applied to any hyperbolic conservation law. The multidimensional Riemann solver is made to be consistent with constraints that emerge naturally from the Galerkin projection of the self-similar states within the wave model. When the full eigenstructure in both directions is used in the present Riemann solver, it becomes a complete Riemann solver in a multidimensional sense. I.e., all the intermediate waves are represented in the multidimensional wave model. The work also presents, for the very first time, an important analysis of the dissipation characteristics of multidimensional Riemann solvers. The present Riemann solver results in the most efficient implementation of a multidimensional Riemann solver with sub-structure. Because it preserves stationary linearly degenerate waves, it might also help with well-balancing. Implementation-related details are presented in pointwise fashion for the one-dimensional HLLI Riemann solver as well as the multidimensional MuSIC Riemann solver. Several stringent test problems drawn from hydrodynamics, MHD and relativistic MHD are presented to show that the method works very well on structured meshes. Our results demonstrate the versatility of our method. The reader is also invited to watch a vide
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