Spectral (finite) volume method for conservation laws on unstructured grids. Basic formulation.

*(English)*Zbl 0997.65115Summary: 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.

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 |

##### Keywords:

spectral volume method; unstructured grid; conservation laws; Riemann solvers; total variation diminishing; total variation bounded; spurious oscillations##### Software:

SHASTA
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\textit{Z. J. Wang}, J. Comput. Phys. 178, No. 1, 210--251 (2002; Zbl 0997.65115)

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