Derivative-Based Finite-Volume MR-HWENO Scheme for Steady-State Problems (Communications in Computational Physics, 2024)
In this paper, we further extend the derivative-based finite-volume multi-resolution Hermite weighted essentially non-oscillatory (MR-HWENO) scheme proposed in our previous article (Li, Shu and Qiu, J. Comput. Phys., 446:110653, 2021) tosimulate the steady-state problem. When dealing with the steady-state problem, theprocess of updating and reconstructing the function values is similar to the previousscheme, but the treatment of the derivative values is changed. To be more specific, instead of evolving in time, in the sense of cell averages, the scheme uses the derivativeat the current time step and the function at the next time step to reconstruct the derivative at the next time step by direct linear interpolation. There are two advantages forthis approach: the first is its high efficiency, when handling the derivative, neither theupdate on time nor the calculation of nonlinear weights is required; in the meantime,the CFL number can still be taken up to 0.6 as in the original scheme; the second isits strong convergence, the corresponding average residual can quickly converge tomachine accuracy, thus obtaining the desired steady-state solution. One- and two-dimensional numerical experiments are given to verify the high efficiency and strongconvergence of the proposed MR-HWENO scheme for the steady-state problems.
Keywords: Finite-volume; multi-resolution scheme; HWENO scheme; Runge-Kutta method; steady-state problem.