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Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids. (English) Zbl 1185.65150
The paper deals with with an efficient and accurate limiting strategy for multi-dimensional hyperbolic conservation laws on unstructured grids. The multidimensional limiting process (MLP) which has been successfully proposed on structured grids is extended to unstructured grids. The MLP condition can guarantee a higher-order spatial accuracy and improved convergence without yielding spurious oscillations. The reconstruction procedure on unstructured grids imposes an additional difficulty. Interpolation on unstructured grids is not readily derived by difference formula due to random indexing. Following the MUSCL-type framework, reconstruction methods were mainly focused on the calculation of the solution gradient. The authors consider the multi-dimensional hyperbolic conservation law \[ \frac{\partial \mathbf{Q}}{\partial t}+\nabla \cdot \mathbf{F} =0, \] \(\mathbf{Q}\) being the variable state vector and \(\mathbf{F}\) the flux vector. Depending on the location of the physical variables, one may use a cell-centered or a cell-vortex method. Here the first approach is used. In the next section the multi-dimensional limiting process is considered. The last paragraph contains numerical results. The following examples are presented: \[ q_{t}+\mathbf{a}\nabla q=0, \] where \(\mathbf{a}\) is the constant flux velocity, and the rotational flow
\[ q_{t}+ \mathbf{a}\nabla q=0 \qquad \mathbf{a}=(-(y-0.5),(x-0.5)). \]

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
76M20 Finite difference methods applied to problems in fluid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
Full Text: DOI
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