On the convection boundedness of numerical schemes across discontinuities

Abstract

This short note introduces a novel diagnostic tool for evaluating the convection boundedness properties of numerical schemes across discontinuities. The proposed method is based on the convection boundedness criterion and the normalised variable diagram. By utilising this tool, we can determine the CFL conditions for numerical schemes to satisfy the convection boundedness criterion, identify the locations of over- and under-shoots, optimize the free parameters in the schemes, and develop strategies to prevent numerical oscillations across the discontinuity. We apply the diagnostic tool to assess representative discontinuity-capturing schemes, including THINC, fifth-order WENO, and fifth-order TENO, and validate the conclusions drawn through numerical tests. We further demonstrate the application of the proposed method by formulating a new THINC scheme with less stringent CFL conditions.

Figures (5)

• Figure 1: The variation of $\tilde{\phi}_{i+1/2}$ with $\tilde{\phi}_i$ for the THINC scheme across the discontinuity with different $\beta$ value.
• Figure 2: Numerical solutions for the advection of a square wave using the THINC scheme with different $\beta$ values at various CFL numbers.
• Figure 3: The normalised variable diagram for the WENOJS5, WENOZ5 and TENO5 schemes across the discontinuity.
• Figure 4: Numerical solutions for the advection of a square wave using the WENO and TENO scheme with different parameters at various CFL numbers.
• Figure 5: Comparisons between the original THINC scheme and the new THINC scheme which is less constrained by the CFL number. (a) 1D advection of a square wave. (b)(c) 2D Zalesak test.

Theorems & Definitions (1)

• proof