SCOUT: Semi-Lagrangian COnservative and Unconditionally sTable schemes for nonlinear advection-diffusion problems

TL;DR

SCOUT develops a conservative semi-Lagrangian scheme for nonlinear advection-diffusion by integrating the governing equations over space-time control volumes bounded by backward characteristics. Fluxes are computed via Gauss theorem, with nonlinear feet solved by Newton iterations when needed, and diffusion is incorporated through a characteristic-based Crank-Nicolson discretization that yields unconditional stability and second-order accuracy. A broad suite of linear and nonlinear advection and advection-diffusion tests demonstrates strict mass conservation, robustness at large CFL numbers, and accurate shock and diffusion handling. The approach advances semi-Lagrangian methods by tightly embedding conservation and diffusion treatment within the characteristic framework, enabling stable large-time stepping for challenging nonlinear problems.

Abstract

In this work, we propose a new semi-Lagrangian (SL) finite difference scheme for nonlinear advection-diffusion problems. To ensure conservation, which is fundamental for achieving physically consistent solutions, the governing equations are integrated over a space-time control volume constructed along the characteristic curves originating from each computational point. By applying Gauss theorem, all space-time surface integrals can be evaluated. For nonlinear problems, a nonlinear equation must be solved to find the foot of the characteristic, while this is not needed in linear cases. This formulation yields SL schemes that are fully conservative and unconditionally stable, as verified by numerical experiments with CFL numbers up to 100. Moreover, the diffusion terms are, for the first time, directly incorporated within a conservative semi-Lagrangian framework, leading to the development of a novel characteristic-based Crank-Nicolson discretization in which the diffusion contribution is implicitly evaluated at the foot of the characteristic. A broad set of benchmark tests demonstrates the accuracy, robustness, and strict conservation property of the proposed method, as well as its unconditional stability.

Figures (11)

• Figure 1: Space-time domain $\Omega_L$ used to derive the conservative $\text{SL}$ scheme in the linear case with constant velocity.
• Figure 2: Constant linear advection case: comparison between the solution at the final time provided by the classical $\text{SL}$ scheme, the novel conservative $\text{SCOUT}$ scheme, and the exact one. Left: Test 1. Right: Test 2. In both tests, we set $N_x=100$, $T=1$, $\nu=0$ and $\text{CFL}=10$.
• Figure 3: Variable coefficient case. Left: comparison among the $\text{SCOUT}$ scheme, the $\text{SL}$ scheme and the exact solution. Right: plot of the number of cell interfaces crossed by each trajectory during the first time step of the computation. Results are obtained by setting $N_x=200$, $T=1.5$, $\nu=0$, $\text{CFL}=20$.
• Figure 4: Burgers' equation with analytical solution: Tavelli test. Left: comparison among the $\text{SCOUT}$ scheme, the $\text{SL}$ scheme and the exact solution.Right: plot of the number of cell interfaces crossed by each trajectory during the first time step of the computation. Results are obtained by setting $N_x=100$, $T=0.2$, $\nu=0$, $\text{CFL}=10$.
• Figure 5: Numerical solutions for the nonlinear Burgers' equation with initial data $q_0(x)=\sqrt{2}/2+\sin(\pi x)$ computed on $N_x=100$ cells, with $\text{CFL}=10$. Top left: $T=0.7/\pi$. Top right: $T=1.3/\pi$. Bottom: $T=2/\pi$. In the first case the solution is still smooth and no artificial viscosity is added. In the other two cases, we make a comparison with a reference solution computed on $N_x=800$ cells, considering also an additional artificial viscosity with coefficient $\nu=5.0e-3$.