Further Studies on the Self-Adaptive Theta Scheme for Conservation Laws

Abstract The finite volume, self-adaptive theta (SATh) scheme was defined in Arbogast and Huang, A self-adaptive theta scheme using discontinuity aware quadrature for solving conservation laws , IMA J. Numer. Anal. (2022). The basic scheme evolves both the local space and space-time averages of the solution in time with an implicitly defined theta parameter. Here, the scheme is extended to unstructured meshes in multiple space dimensions, general numerical flux functions, and higher (formally second) order using WENO reconstructions. Theoretical results apply to the one space dimension, upstream weighted case, in the setting of a monotone solution. In this case, if the theta parameter is bounded below by $$\theta _{\min }=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>θ</mml:mi> <mml:mo>min</mml:mo> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , it is shown that SATh is stable, L-stable for the linear problem, total variation diminishing (TVD), and maximum principle preserving (MPP). These results generalize those known previously with the assumption that $$\theta _{\min }=1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>θ</mml:mi> <mml:mo>min</mml:mo> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . Numerical tests for problems with contact discontinuities, shocks, and rarefactions show that SATh performs better than finite volume schemes using backward Euler time stepping. Moreover, SATh gives solutions about as sharp as when using Crank-Nicolson time stepping, but SATh is non-oscillatory. In cases covered by the theoretical results, SATh combined with a Lax-Friedrichs numerical flux (rather than upstream weighting) appears to be TVD and MPP. SATh is non-oscillatory if $$\theta _{\min }=1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>θ</mml:mi> <mml:mo>min</mml:mo> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , but if $$\theta _{\min }=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>θ</mml:mi> <mml:mo>min</mml:mo> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and the solution is not monotone, it can develop oscillations. The higher order SATh scheme converges to order two and compares favorably with CN, but is less oscillatory.