High order accurate semi-implicit WENO schemes for hyperbolic balance laws

• Published in: Applied mathematics and computation 2011-07, Vol.217 (21), p.8611-8629
• Main Authors: CRNJARIC-ZIC, Nelida, CRNKOVIC, Bojan
• Format: Article
• Language: English
• Subjects: Combinatorics; Combinatorics. Ordered structures; Conservation; Conservation law; Designs and configurations; Discretization; Exact sciences and technology; Hyperbolic balance law; Implicit schemes; Laws; Mathematical analysis; Mathematical models; Mathematics; Measure and integration; Numerical analysis; Numerical analysis. Scientific computation; Open channel flow; Ordinary differential equations; Partial differential equations; Sciences and techniques of general use; Stability; Strong stability preserving methods; Time integration; Well-balanced schemes; WENO schemes
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2011.03.098

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89–112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge–Kutta methods, Appl. Numer. Math. 58 (2008) 1675–1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed. A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.