2,075 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
Assessment of a high-resolution central scheme for the solution of the relativistic hydrodynamics equations
We assess the suitability of a recent high-resolution central scheme
developed by Kurganov & Tadmor (2000) for the solution of the relativistic
hydrodynamics equations. The novelty of this approach relies on the absence of
Riemann solvers in the solution procedure. The computations we present are
performed in one and two spatial dimensions in Minkowski spacetime. Standard
numerical experiments such as shock tubes and the relativistic flat-faced step
test are performed. As an astrophysical application the article includes
two-dimensional simulations of the propagation of relativistic jets using both
Cartesian and cylindrical coordinates. The simulations reported clearly show
the capabilities of the numerical scheme to yield satisfactory results, with an
accuracy comparable to that obtained by the so-called high-resolution
shock-capturing schemes based upon Riemann solvers (Godunov-type schemes), even
well inside the ultrarelativistic regime. Such central scheme can be
straightforwardly applied to hyperbolic systems of conservation laws for which
the characteristic structure is not explicitly known, or in cases where the
exact solution of the Riemann problem is prohibitively expensive to compute
numerically. Finally, we present comparisons with results obtained using
various Godunov-type schemes as well as with those obtained using other
high-resolution central schemes which have recently been reported in the
literature.Comment: 14 pages, 12 figures, to appear in A&
An HLLC Riemann Solver for Relativistic Flows: I. Hydrodynamics
We present an extension of the HLLC approximate Riemann solver by Toro,
Spruce and Speares to the relativistic equations of fluid dynamics. The solver
retains the simplicity of the original two-wave formulation proposed by Harten,
Lax and van Leer (HLL) but it restores the missing contact wave in the solution
of the Riemann problem. The resulting numerical scheme is computationally
efficient, robust and positively conservative. The performance of the new
solver is evaluated through numerical testing in one and two dimensions.Comment: 12 pages, 12 figure
High-Order Unstructured Lagrangian One-Step WENO Finite Volume Schemes for Non-Conservative Hyperbolic Systems: Applications to Compressible Multi-Phase Flows
In this article we present the first better than second order accurate
unstructured Lagrangian-type one-step WENO finite volume scheme for the
solution of hyperbolic partial differential equations with non-conservative
products. The method achieves high order of accuracy in space together with
essentially non-oscillatory behavior using a nonlinear WENO reconstruction
operator on unstructured triangular meshes. High order accuracy in time is
obtained via a local Lagrangian space-time Galerkin predictor method that
evolves the spatial reconstruction polynomials in time within each element. The
final one-step finite volume scheme is derived by integration over a moving
space-time control volume, where the non-conservative products are treated by a
path-conservative approach that defines the jump terms on the element
boundaries. The entire method is formulated as an Arbitrary-Lagrangian-Eulerian
(ALE) method, where the mesh velocity can be chosen independently of the fluid
velocity.
The new scheme is applied to the full seven-equation Baer-Nunziato model of
compressible multi-phase flows in two space dimensions. The use of a Lagrangian
approach allows an excellent resolution of the solid contact and the resolution
of jumps in the volume fraction. The high order of accuracy of the scheme in
space and time is confirmed via a numerical convergence study. Finally, the
proposed method is also applied to a reduced version of the compressible
Baer-Nunziato model for the simulation of free surface water waves in moving
domains. In particular, the phenomenon of sloshing is studied in a moving water
tank and comparisons with experimental data are provided
Relativistic Hydrodynamic Flows Using Spatial and Temporal Adaptive Structured Mesh Refinement
Astrophysical relativistic flow problems require high resolution
three-dimensional numerical simulations. In this paper, we describe a new
parallel three-dimensional code for simulations of special relativistic
hydrodynamics (SRHD) using both spatially and temporally structured adaptive
mesh refinement (AMR). We used the method of lines to discretize the SRHD
equations spatially and a total variation diminishing (TVD) Runge-Kutta scheme
for time integration. For spatial reconstruction, we have implemented piecewise
linear method (PLM), piecewise parabolic method (PPM), third order convex
essentially non-oscillatory (CENO) and third and fifth order weighted
essentially non-oscillatory (WENO) schemes. Flux is computed using either
direct flux reconstruction or approximate Riemann solvers including HLL,
modified Marquina flux, local Lax-Friedrichs flux formulas and HLLC. The AMR
part of the code is built on top of the cosmological Eulerian AMR code {\sl
enzo}. We discuss the coupling of the AMR framework with the relativistic
solvers. Via various test problems, we emphasize the importance of resolution
studies in relativistic flow simulations because extremely high resolution is
required especially when shear flows are present in the problem. We also
present the results of two 3d simulations of astrophysical jets: AGN jets and
GRB jets. Resolution study of those two cases further highlights the need of
high resolutions to calculate accurately relativistic flow problems.Comment: 14 pages, 23 figures. A section on 3D GRB jet simulation added.
Accepted by ApJ
A multidimensional grid-adaptive relativistic magnetofluid code
A robust second order, shock-capturing numerical scheme for multi-dimensional
special relativistic magnetohydrodynamics on computational domains with
adaptive mesh refinement is presented. The base solver is a total variation
diminishing Lax-Friedrichs scheme in a finite volume setting and is combined
with a diffusive approach for controlling magnetic monopole errors. The
consistency between the primitive and conservative variables is ensured at all
limited reconstructions and the spatial part of the four velocity is used as a
primitive variable. Demonstrative relativistic examples are shown to validate
the implementation. We recover known exact solutions to relativistic MHD
Riemann problems, and simulate the shock-dominated long term evolution of
Lorentz factor 7 vortical flows distorting magnetic island chains.Comment: accepted for publication in Computer Physics Communication
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
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