207 research outputs found
ADER-WENO Finite Volume Schemes with Space-Time Adaptive Mesh Refinement
We present the first high order one-step ADER-WENO 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 one-step
time discretization is achieved using a local space-time discontinuous Galerkin
predictor method. Due to the one-step nature of the underlying scheme, the
resulting algorithm is particularly well suited for an AMR strategy on
space-time adaptive meshes, i.e.with time-accurate local time stepping. The AMR
property has been implemented 'cell-by-cell', with a standard tree-type
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
speed-up 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.Comment: With updated bibliography informatio
High Order Cell-Centered Lagrangian-Type Finite Volume Schemes with Time-Accurate Local Time Stepping on Unstructured Triangular Meshes
We present a novel cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE)
finite volume scheme on unstructured triangular meshes that is high order
accurate in space and time and that also allows for time-accurate local time
stepping (LTS). The new scheme uses the following basic ingredients: a high
order WENO reconstruction in space on unstructured meshes, an element-local
high-order accurate space-time Galerkin predictor that performs the time
evolution of the reconstructed polynomials within each element, the computation
of numerical ALE fluxes at the moving element interfaces through approximate
Riemann solvers, and a one-step finite volume scheme for the time update which
is directly based on the integral form of the conservation equations in
space-time. The inclusion of the LTS algorithm requires a number of crucial
extensions, such as a proper scheduling criterion for the time update of each
element and for each node; a virtual projection of the elements contained in
the reconstruction stencils of the element that has to perform the WENO
reconstruction; and the proper computation of the fluxes through the space-time
boundary surfaces that will inevitably contain hanging nodes in time due to the
LTS algorithm. We have validated our new unstructured Lagrangian LTS approach
over a wide sample of test cases solving the Euler equations of compressible
gasdynamics in two space dimensions, including shock tube problems, cylindrical
explosion problems, as well as specific tests typically adopted in Lagrangian
calculations, such as the Kidder and the Saltzman problem. When compared to the
traditional global time stepping (GTS) method, the newly proposed LTS algorithm
allows to reduce the number of element updates in a given simulation by a
factor that may depend on the complexity of the dynamics, but which can be as
large as 4.7.Comment: 31 pages, 13 figure
High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes
We present a new family of very high order accurate direct
Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous
Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on
moving 2D Voronoi meshes that are regenerated at each time step and which
explicitly allow topology changes in time.
The Voronoi tessellations are obtained from a set of generator points that
move with the local fluid velocity. We employ an AREPO-type approach, which
rapidly rebuilds a new high quality mesh rearranging the element shapes and
neighbors in order to guarantee a robust mesh evolution even for vortex flows
and very long simulation times. The old and new Voronoi elements associated to
the same generator are connected to construct closed space--time control
volumes, whose bottom and top faces may be polygons with a different number of
sides. We also incorporate degenerate space--time sliver elements, needed to
fill the space--time holes that arise because of topology changes. The final
ALE FV-DG scheme is obtained by a redesign of the fully discrete direct ALE
schemes of Boscheri and Dumbser, extended here to moving Voronoi meshes and
space--time sliver elements. Our new numerical scheme is based on the
integration over arbitrary shaped closed space--time control volumes combined
with a fully-discrete space--time conservation formulation of the governing PDE
system. In this way the discrete solution is conservative and satisfies the GCL
by construction.
Numerical convergence studies as well as a large set of benchmarks for
hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and
robustness of the proposed method. Our numerical results clearly show that the
new combination of very high order schemes with regenerated meshes with
topology changes lead to substantial improvements compared to direct ALE
methods on conforming meshes
Very High Order \PNM Schemes on Unstructured Meshes for the Resistive Relativistic MHD Equations
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 \PNM schemes as introduced in \cite{Dumbser2008}
for hyperbolic conservation laws and a high order accurate unsplit time
discretization is achieved using the element-local space-time discontinuous
Galerkin approach proposed in \cite{DumbserEnauxToro} for one-dimensional
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. \cite{Dedneretal}. 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 \PNM schemes from third to fifth order of accuracy
in space and time. We also present some applications with shock waves such as a
classical shock tube problem with different values for the conductivity as well
as a relativistic MHD rotor problem and the relativistic equivalent of the
Orszag-Tang vortex problem. We have verified that the proposed method can
handle equally well the resistive regime and the stiff limit of ideal
relativistic MHD. For these reasons it provides a powerful tool for
relativistic astrophysical simulations involving the appearance of magnetic
reconnection.Comment: 24 pages, 6 figures, submitted to JC
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 Finite Difference WENO Schemes with Constrained Transport for Ideal Magnetohydrodynamic Equations
In this paper, we utilize the maximum-principle-preserving flux limiting
technique, originally designed for high order weighted essentially
non-oscillatory (WENO) methods for scalar hyperbolic conservation laws, to
develop a class of high order positivity-preserving finite difference WENO
methods for the ideal magnetohydrodynamic (MHD) equations. Our schemes, under
the constrained transport (CT) framework, can achieve high order accuracy, a
discrete divergence-free condition and positivity of the numerical solution
simultaneously. Numerical examples in 1D, 2D and 3D are provided to demonstrate
the performance of the proposed method.Comment: 21 pages, 28 figure
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