2,364 research outputs found
Third-order Limiting for Hyperbolic Conservation Laws applied to Adaptive Mesh Refinement and Non-Uniform 2D Grids
In this paper we extend the recently developed third-order limiter function
[J. Sci. Comput., (2016), 68(2), pp.~624--652] to make it
applicable for more elaborate test cases in the context of finite volume
schemes. This work covers the generalization to non-uniform grids in one and
two space dimensions, as well as two-dimensional Cartesian grids with adaptive
mesh refinement (AMR). The extension to 2D is obtained by the common approach
of dimensional splitting. In order to apply this technique without loss of
third-order accuracy, the order-fix developed by Buchm\"uller and Helzel [J.
Sci. Comput., (2014), 61(2), pp.~343--368] is incorporated into the scheme.
Several numerical examples on different grid configurations show that the
limiter function maintains the optimal third-order
accuracy on smooth profiles and avoids oscillations in case of discontinuous
solutions
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 Energy Stable WENO Schemes
A new third-order Energy Stable Weighted Essentially NonOscillatory (ESWENO) finite difference scheme for scalar and vector linear hyperbolic equations with piecewise continuous initial conditions is developed. The new scheme is proven to be stable in the energy norm for both continuous and discontinuous solutions. In contrast to the existing high-resolution shock-capturing schemes, no assumption that the reconstruction should be total variation bounded (TVB) is explicitly required to prove stability of the new scheme. A rigorous truncation error analysis is presented showing that the accuracy of the 3rd-order ESWENO scheme is drastically improved if the tuning parameters of the weight functions satisfy certain criteria. Numerical results show that the new ESWENO scheme is stable and significantly outperforms the conventional third-order WENO finite difference scheme of Jiang and Shu in terms of accuracy, while providing essentially nonoscillatory solutions near strong discontinuities
Relativistic MHD with Adaptive Mesh Refinement
This paper presents a new computer code to solve the general relativistic
magnetohydrodynamics (GRMHD) equations using distributed parallel adaptive mesh
refinement (AMR). The fluid equations are solved using a finite difference
Convex ENO method (CENO) in 3+1 dimensions, and the AMR is Berger-Oliger.
Hyperbolic divergence cleaning is used to control the
constraint. We present results from three flat space tests, and examine the
accretion of a fluid onto a Schwarzschild black hole, reproducing the Michel
solution. The AMR simulations substantially improve performance while
reproducing the resolution equivalent unigrid simulation results. Finally, we
discuss strong scaling results for parallel unigrid and AMR runs.Comment: 24 pages, 14 figures, 3 table
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