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Spectral (finite) volume method for conservation laws on unstructured grids. Basic formulation. (English) Zbl 0997.65115
Summary: A high-order, conservative, yet efficient method named the spectral volume (SV) method is developed for conservation laws on unstructured grids. The concept of a “spectral volume” is introduced to achieve high-order accuracy in an efficient manner similar to spectral element and multidomain spectral methods. Each spectral volume is further subdivided into control volumes, and cell-averaged data from these control volumes are used to reconstruct a high-order approximation in the spectral volume. Then Riemann solvers are used to compute the fluxes at spectral volume boundaries. Cell-averaged state variables in the control volumes are updated independently.
Furthermore, total variation diminishing and total variation bounded limiters are introduced in the SV method to remove/reduce spurious oscillations near discontinuities. Unlike spectral element and multidomain spectral methods, the SV method can be applied to fully unstructured grids. A very desirable feature of the SV method is that the reconstruction is carried out analytically, and the reconstruction stencil is always nonsingular, in contrast to the memory and CPU-intensive reconstruction in a high-order $$k$$-exact finite volume method. Fundamental properties of the SV method are studied and high-order accuracy is demonstrated for several model problems with and without discontinuities.

##### MSC:
 65M70 Spectral, collocation and related 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
SHASTA
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