TVD-MOOD schemes based on implicit-explicit time integration

Abstract

The context of this work is the development of first order total variation diminishing (TVD) implicit-explicit (IMEX) Runge-Kutta (RK) schemes as a basis of a Multidimensional Optimal Order detection (MOOD) approach to approximate the solution of hyperbolic multi-scale equations. A key feature of our newly proposed TVD schemes is that the resulting CFL condition does not depend on the fast waves of the considered model, as long as they are integrated implicitly. However, a result from Gottlieb et al. gives a first order barrier for unconditionally stable implicit TVD-RK schemes and TVD-IMEX-RK schemes with scale-independent CFL conditions. Therefore, the goal of this work is to consistently improve the resolution of a first-order IMEX-RK scheme, while retaining its $L^\infty$ stability and TVD properties. In this work we present a novel approach based on a convex combination between a first-order TVD IMEX Euler scheme and a potentially oscillatory high-order IMEX-RK scheme. We derive and analyse the TVD property for a scalar multi-scale equation and numerically assess the performance of our TVD schemes compared to standard $L$-stable and SSP IMEX RK schemes from the literature. Finally, the resulting TVD-MOOD schemes are applied to the isentropic Euler equations.

Figures (11)

• Figure 1: Approximation of a discontinuous solution with $\Delta x = 0.1$ using the first-order, second-order ARS(2,2,2) and third-order ARS(2,3,3) scheme for $\varepsilon = 1$ (left) and $\varepsilon = 10^{-3}$ (right), with an upwind space discretisation.
• Figure 2: CPU time (in milliseconds) with respect to the parameter $\alpha$, using $\gamma = \gamma^{\text{opt}} = \frac{2}{3}$, in the context of the test case presented in Section \ref{['sec:choice_of_gamma_third_order']}.
• Figure 3: $L^\infty$-error with respect to the parameter $\alpha$, using $\gamma = \gamma^{\text{opt}} = \frac{2}{3}$, $\theta_3 = \frac{3}{8}$ and $\theta_4$, $\lambda$ given by relation \ref{['eq:theta4_lambda_wrt_alpha']}. in the context of the test case presented in Section \ref{['sec:choice_of_gamma_third_order']}. For $\alpha \in (0,0.35)$, the top right panel contains a zoom on the CPU time (data from Figure \ref{['fig:CPU_time_wrt_alpha_third_order']}) and the bottom right panel contains a zoom on the $L^\infty$-error (data from left panel).
• Figure 4: Discontinuous solution \ref{['eq:discontinuous_exact_solution']} of the linear advection problem at time $T_f = 1$ with $\Delta x = 0.1$ for $\varepsilon = 1$ (top) and $\varepsilon = 10^{-3}$ (bottom). The dashed lines denote the acoustic CFL condition \ref{['eq:def_nu_ac']} with $\nu_{\text{ac}} = 0.5$, and the solid lines use the material CFL condition \ref{['eq:def_nu_mat']} with $\nu_{\text{mat}} = 0.5$.
• Figure 5: Error lines in $L^1$ norm (left) and ${L^1_{\text{o}}}$ quasinorm (right) for the discontinuous solution \ref{['eq:discontinuous_exact_solution']} with $\varepsilon = 1$ (top) and $\varepsilon = 10^{-3}$ (bottom).

Theorems & Definitions (6)

• Lemma 1: $L^\infty$ stability, TVD property
• Theorem 2
• Remark 1
• Remark 2
• Remark 3
• Theorem 3