Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids. Basic Formulation

• Published in: Journal of computational physics Vol. 178; no. 1; pp. 210 - 251
• Main Author: Wang, Z.J.
• Format: Journal Article
• Language: English
• Published: 01-05-2002
• ISSN: 0021-9991
• DOI: 10.1006/jcph.2002.7041

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 mull\domain 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. (Author)