Multi-dimensional Limiting Process and Maximum Principle for the Computations of Hyperbolic Conservation Laws on Unstructured Grids

The present paper presents an efficient and accurate limiting strategy for the multidimensional hyperbolic conservation laws on unstructured grids within the framework of finite volume method. The basic idea is to control the distribution of both cell-centered and cell-vertex physical properties to mimic a multi-dimensional nature of flow physics, which can be formulated as so called MLP condition. Mathematically, this condition satisfies the maximum principle which is a complementary condition ensuring monotonicity. Various numerical results show that the MLP is quite effective and accurate in preventing unwanted oscillations and capturing multi-dimensional flow features.This research was supported by NSL (National Space Lab.) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (Grant 20090091724), the second stage of the Brain Korea 21 Project for Mechanical and Aerospace Engineering Research at Seoul National University, and by Agency for Defense Development.OAIID:oai:osos.snu.ac.kr:snu2010-01/104/0000004648/17SEQ:17PERF_CD:SNU2010-01EVAL_ITEM_CD:104USER_ID:0000004648ADJUST_YN:NEMP_ID:A001138DEPT_CD:446CITE_RATE:0FILENAME:Multi-dimensional_Limiting_Process_and_Maximum_Principle.pdfDEPT_NM:기계항공공학부EMAIL:[email protected]:

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

Adaptive multiresolution computations applied to detonations

A space-time adaptive method is presented for the reactive Euler equations describing chemically reacting gas flow where a two species model is used for the chemistry. The governing equations are discretized with a finite volume method and dynamic space adaptivity is introduced using multiresolution analysis. A time splitting method of Strang is applied to be able to consider stiff problems while keeping the method explicit. For time adaptivity an improved Runge--Kutta--Fehlberg scheme is used. Applications deal with detonation problems in one and two space dimensions. A comparison of the adaptive scheme with reference computations on a regular grid allow to assess the accuracy and the computational efficiency, in terms of CPU time and memory requirements.Comment: Zeitschrift f\"ur Physicalische Chemie, accepte