Mechanism and Improvement of the Harten-Lax-van Leer Scheme for All-Speed Flows

TL;DR

This paper analyzes the mechanisms of classical and preconditioned Harten-Lax-van Leer schemes for all-speed flows, identifies their issues, and proposes a new scheme that overcomes these problems by modifying the momentum-derivative term based on local Mach number.

Contribution

It introduces a novel all-speed HLL scheme that addresses accuracy, global cut-off, and checkerboard problems using a simple modification based on the local Mach number.

Findings

The classical HLL scheme's accuracy is limited by a large velocity-derivative coefficient.

The preconditioned HLL scheme suffers from global cut-off and pressure-velocity decoupling issues.

The proposed scheme effectively overcomes these problems, improving all-speed flow simulations.

Abstract

Based on the three rules developed from the Roe-type scheme, the mechanisms of the classical and preconditioned Harten-Lax-van Leer (HLL) schemes are analyzed. For the classical HLL scheme, the accuracy problem is attributable to the extremely large coefficient of the velocity-derivative term of the momentum equation. For the preconditioned HLL scheme, the global cut-off problem is attributable to the denominator of the coefficients, whereas the particular pressure gradient sensor problem results from the loss of the capability to suppress the checkerboard pressure-velocity decoupling. A new all-speed HLL scheme, which can overcome these problems by only multiplying the momentum-derivative term in the momentum equation by a function related to the local Mach number, is proposed. More importantly, the present study shows the generality of the three rules, which can be powerful tools for analyzing and proposing schemes. The second rule involving the checkerboard problem is also improved by emphasizing that the coefficients of pressure-derivative term in the continuity and energy equations should be changed simultaneously.