Efficient Alternative Finite Difference WENO Schemes for Hyperbolic Systems with Non-Conservative Products
<|reference_start|>Efficient Alternative Finite Difference WENO Schemes for Hyperbolic Systems with Non-Conservative Products: Higher order finite difference Weighted Essentially Non-Oscillatory (WENO) schemes for conservation laws represent a technology that has been reasonably consolidated. They are extremely popular because, when applied to multidimensional problems, they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes. They come in two flavors. There is the classical finite difference WENO (FD-WENO) method (Shu and Osher, J. Comput. Phys., 83 (1989) 32-78). However, in recent years there is also an alternative finite difference WENO (AFD-WENO) method which has recently been formalized into a very useful general-purpose algorithm for conservation laws (Balsara et al., Efficient Alternative Finite Difference WENO Schemes for Hyperbolic Conservation Laws, submitted to CAMC (2023)). However, the FD-WENO algorithm has only very recently been formulated for hyperbolic systems with non-conservative products (Balsara et al., Efficient Finite Difference WENO Scheme for Hyperbolic Systems with Non-Conservative Products, to appear CAMC (2023)). In this paper we show that there are substantial advantages in obtaining an AFD-WENO algorithm for hyperbolic systems with non-conservative products. Such an algorithm is documented in this paper. We present an AFD-WENO formulation in fluctuation form that is carefully engineered to retrieve the flux form when that is warranted and nevertheless extends to non-conservative products. The method is flexible because it allows any Riemann solver to be used. The formulation we arrive at is such that when non-conservative products are absent it reverts exactly to the formulation in the second citation above which is in exact flux conservation form. The ability to transition to a precise conservation form when non-conservative products are absent ensures, via the Lax-Wendroff theorem, that shock locations will be exactly ...<|reference_end|>
@article{balsara2024efficient,
    title={Efficient Alternative Finite Difference WENO Schemes for Hyperbolic
  Systems with Non-Conservative Products},
    author={Dinshaw S. Balsara, Deepak Bhoriya, Chi-Wang Shu and Harish Kumar},
    journal={arXiv preprint arXiv:2403.01266},
    year={2024},
    doi={10.1007/s42967-024-00374-1},
    archivePrefix={arXiv},
    eprint={2403.01266},
    primaryClass={math.NA cs.NA math-ph math.MP physics.flu-dyn}
}