Commun. Comput. Phys., 1 (2006), pp. 548-574.

Two-Dimensional Multi-Domain Hybrid Spectral-WENO Methods for Conservation Laws

Bruno Costa 1, Wai Sun Don 2*, David Gottlieb 2, Radislav Sendersky 2

1 Departamento de Matematica Aplicada, IM-UFRJ, Caixa Postal 68530, Rio de Janeiro, RJ, C.E.P. 21945-970, Brazil.
2 Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, USA.

Received 14 November 2005; Accepted (in revised version) 16 January 2006
Communicated by Pingwen Zhang


The multi-domain hybrid Spectral-WENO method (Hybrid) is introduced for the numerical solution of two-dimensional nonlinear hyperbolic systems in a Cartesian physical domain which is partitioned into a grid of rectangular subdomains. The main idea of the Hybrid scheme is to conjugate the spectral and WENO methods for solving problems with shock or high gradients such that the scheme adapts its solver spatially and temporally depending on the smoothness of the solution in a given subdomain. Built as a multi-domain method, an adaptive algorithm is used to keep the solutions parts exhibiting high gradients and discontinuities always inside WENO subdomains while the smooth parts of the solution are kept inside spectral ones, avoiding oscillations related to the well-known Gibbs phenomenon and increasing the numerical efficiency of the overall scheme. A higher order version of the multi-resolution analysis proposed by Harten is used to determine the smoothness of the solution in each subdomain. We also discuss interface conditions for the two-dimensional problem and the switching procedure between WENO and spectral subdomains. The Hybrid method is applied to the two-dimensional Shock-Vortex Interaction and the Richtmyer-Meshkov Instability (RMI) problems.

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Key words: Spectral; WENO; multi-resolution; multi-domain; hybrid; conservation laws.

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Correspondence to: Wai Sun Don , Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, USA. Email:

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