This paper is concerned with the development of high order methods for the numerical approximation of one-dimensional nonconservative hyperbolic systems. In particular, we are interested in high order extensions of the generalized Roe methods introduced by I. Toumi in 1992, based on WENO reconstruction of states. We also investigate the well-balanced properties of the resulting schemes. Finally...

First, a new reconstruction strategy is proposed to improve the accuracy of fifth-order weighted essentially non-oscillatory (WENO) scheme. It has been noted that conventional WENO schemes still suffer from excessive numerical dissipation near-critical regions. One reasons they tend under-use all adjacent smooth substencils thus fail realize optimal interpolation. Hence in this work, modified (...

The ADER approach is the extended Godunov-type schemes[1] which construct nonoscillatory explicit one-step schemes with very high order of accuracy in space and time by solving the DRPs (derivative Riemann problems). The ADER approach has been developed from the schemes for the linear scalar hyperbolic equations[2] to those for nonlinear multi-dimensional systems[3,4,5,6,7], and further to thos...

In this paper, we develop a class of nonlinear compact schemes based on our previous linear central compact schemes with spectral-like resolution [X. Liu, S. Zhang, H. Zhang and C-W. Shu, A new class of central compact schemes with spectral-like resolution I: Linear schemes, Journal of Computational Physics 248 (2013) 235-256]. In our approach, we compute the flux derivatives on the cell-nodes ...

The purpose of this paper is to carry out a modification of the finite volume WENO (weighted essentially non-oscillatory) scheme of Titarev and Toro [10]. This modification is done by using the third order TVD flux [10] as building blocks in spatially fifth order WENO schemes, instead of the second order TVD flux proposed by Titarev and Toro. The resulting scheme improves both the original and ...

We consider implicit-explicit (IMEX) Runge Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strong-stabilitypreserving (SSP) scheme, and the implicit part is treated by an L-stable diagonally implicit Runge Kutta (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space...

Numerical methods with different orders of accuracy are proposed to approximate hyperbolic models for chemosensitive movements. On the one hand, firstand second-order wellbalanced finite volume schemes are presented. This approach provides exact conservation of the steady state solutions. On the other hand, a high-order finite difference weighted essentially nonoscillatory (WENO) scheme is cons...