High Order Residual Distribution Conservative Finite Difference HWENO Scheme for Steady State Problems

TL;DR

This paper introduces a high order residual distribution HWENO scheme for steady state conservation laws, offering improved efficiency, compactness, and accuracy through a novel integration approach that avoids auxiliary equations.

Contribution

The paper develops a new high order residual distribution HWENO scheme that simplifies computations and enhances accuracy for steady state problems without auxiliary equations.

Findings

Achieves high order accuracy in scalar and system problems.

Reduces computational storage and CPU time compared to traditional HWENO.

Demonstrates smaller numerical errors and simpler boundary treatment.

Abstract

In this paper, we develop a high order residual distribution (RD) method for solving steady state conservation laws in a novel Hermite weighted essentially non-oscillatory (HWENO) framework recently developed in [24]. In particular, we design a high order HWENO integration for the integrals of source term and fluxes based on the point value of the solution and its spatial derivatives, and the principles of residual distribution schemes are adapted to obtain steady state solutions. Two advantages of the novel HWENO framework have been shown in [24]: first, compared with the traditional HWENO framework, the proposed method does not need to introduce additional auxiliary equations to update the derivatives of the unknown variable, and just compute them from the current point value of the solution and its old spatial derivatives, which saves the computational storage and CPU time, and thereby improve the computational efficiency of the traditional HWENO framework. Second, compared with the traditional WENO method, reconstruction stencil of the HWENO methods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller at the same grid. Thus, it is also a compact scheme when we design the higher order accuracy, compared with that in [11] Chou and Shu proposed. Extensive numerical experiments for one- and two-dimensional scalar and systems problems confirm the high order accuracy and good quality of our scheme.