High order cell-centered Lagrangian-type finite volume schemes with time-accurate local time stepping on unstructured triangular meshes

Abstract

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). It extends our previous investigations on high order Lagrangian finite volume schemes with LTS carried out in [36] in one space dimension. 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.

An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

“…To improve the computational efficiency of our algorithm, we use the MOOD paradigm . In the popular WENO technique has been used for the reconstruction procedure, that in the polynomial formulation presented in implies the use of seven (in 2D) or nine (in 3D) reconstruction stencils, each of them producing a reconstruction polynomial of the form () that has to be blended nonlinearly to obtain the final reconstruction polynomial for each element . On the contrary, the MOOD approach only needs one reconstruction stencil to derive one final reconstruction polynomial , which is used as is, that is, unlimited .…”

Direct Arbitrary-Lagrangian–Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws

“…with Δt V,i denoting the unknown timestep which satisfies the volume criterion (19). By using expression (20) to explicitly derive a formula for the new cell volume |T n+1 i |, the volume restriction (19) constitutes a second or third order algebraic equation for the unknown Δt V,i , in two and three space dimensions, respectively.…”

“…By using expression (20) to explicitly derive a formula for the new cell volume |T n+1 i |, the volume restriction (19) constitutes a second or third order algebraic equation for the unknown Δt V,i , in two and three space dimensions, respectively. The final timestep will be given by taking as always the minimum between all Δt V,i , i.e.…”

“…In order to improve the efficiency of the algorithm, in [15] the use of a genuinely multi-dimensional HLL-type Riemann solver [4,6,9] led to larger timesteps, hence yielding a computationally more efficient scheme compared to a method based on classical one-dimensional Riemann solvers. A second attempt for reducing the computational efforts has been presented in [46,19], where a local time stepping scheme on moving meshes is presented. The local time stepping allows the algorithm to be free from a global CFL stability condition, because each control volume reaches the final time of the simulation using its own optimal timestep.…”

An Efficient Quadrature-Free Formulation for High Order Arbitrary-Lagrangian–Eulerian ADER-WENO Finite Volume Schemes on Unstructured Meshes

“…A first attempt in improving the efficiency of the flux computation has been presented in [8], where the use of a genuinely multi-dimensional HLL-type Riemann solver [1,2,30] yields larger time steps and therefore leads to a computationally more efficient scheme compared to a method based on classical one-dimensional Riemann solvers. Another possibility is given by the adoption of a local time stepping scheme [12,33], that allows each control volume to reach the final time of the simulation using its own optimal timestep, which obeys only a local CFL stability condition instead of a global one.…”