A new flux-vector splitting compact finite volume scheme
A new high resolution finite volume method is proposed here that uses flux-vector splitting as the building block to represent the physical processes. The high accuracy projection in space is obtained here, inter alia, by using some compact schemes of Sengupta et al. [1] and Lele [2] - that have extremely high accuracy in spectral plane. This produces a new class of finite volume (FV) scheme for spatial discretization, that has been analyzed for spectral accuracy, numerical stability and dispersion relation preservation (DRP) property following the full-domain analysis method of [1] using two different time integration strategies. These combination of spatial and temporal methods have been tested for two linear wave problems, reported in [1,3] and the solution of Burgers' equation is compared with the analytical results presented in Adams and Shariff [4]. The shock capturing method is based on the proposed technique in [5], that does not require the usage of any nonlinear differencing techniques or limiters. To show that the proposed method can also handle irregular grids, the Burgers' equation is solved using non-uniform grids. Also to demonstrate the utility of the proposed scheme to solve practical problems, the benchmark problem of nonlinear wave propagation in a one-dimensional shock-tube has been solved with the initial data as given in [23] by solving the Euler equation. The analysis and comparison of this flux vector splitting scheme with other well known methods clearly demonstrate the superior scale resolution of the proposed method, at the same time providing a neutrally stable scheme when used with four stage Runge-Kutta scheme. Also the same scheme displays extended ranges of wave numbers and circular frequencies over which DRP property is valid. The computed results of three test cases and their comparison with analytical results clearly reveal that the presented method can be used for practical problems. &COPY; 2005 Elsevier Inc. All rights reserved.
finite volume method compact schemes nonlinear wave propagation flux vector splitting schemes shock capturing technique