Unconditionally local bounds preserving numerical scheme based on inverse Lax-Wendroff procedure for advection on networks

TL;DR

This paper addresses stable and accurate advection simulation on networks by developing an unconditionally bound-preserving implicit scheme based on the inverse Lax–Wendroff procedure. By exchanging space and time roles, it enables forward substitution, with a space–time limiter that combines second-order accuracy in space and third-order accuracy in time, and enforces the discrete minimum–maximum principle (DMP) unconditionally. A predictor–corrector strategy and limiter-based refinements (including WENO-type variants) handle potential nonlinearities and suppress nonphysical oscillations, making the method robust for linear and nonlinear retardation effects. The approach is validated on several network benchmarks, including a realistic sewer network, demonstrating stable, accurate, and bound-preserving transport with efficient computation and straightforward extensibility to nonlinear coefficients.

Abstract

We derive an implicit numerical scheme for the solution of advection equation where the roles of space and time variables are exchanged using the inverse Lax-Wendroff procedure. The scheme contains a linear weight for which it is always second order accurate in time and space, and the stencil in the implicit part is fully upwinded for any value of the weight, enabling a direct computation of numerical solutions by forward substitution. To fulfill the local bounds for the solution represented by the discrete minimum and maximum principle (DMP), we use a predicted value obtained with the linear weight and check a priori if the DMP is valid. If not, we can use either a nonlinear weight or a limiter function that depends on Courant number and apply such a high-resolution version of the scheme to obtain a corrected value. The advantage of the scheme obtained with the inverse Lax-Wendroff procedure is that only in the case of too small Courant numbers, the limiting is towards the first order accurate scheme, which is not a situation occurring in numerical simulations with implicit schemes very often. In summary, the local bounds are satisfied up to rounding errors unconditionally for any Courant numbers, and the formulas for the predictor and the corrector are explicit. The high-resolution scheme can be extended straightforwardly for advection with nonlinear retardation coefficient with numerical solutions satisfying the DMP, and a scalar nonlinear algebraic equation has to be solved to obtain each predicted and corrected value. In numerical experiments, including transport on a sewer network, we can confirm the advantageous properties of numerical solutions for several representative examples.

Figures (17)

• Figure 1: Schematic diagram of the network using the notation outlined in this section.
• Figure 2: Three dimensional representation of a part of the sewer network in Revúca.
• Figure 3: Comparison of the example in Section \ref{['sec-eoc']}: the four curves correspond to the (interpolated) exact solution (orange), the 3rd order method (red), the high-resolution method (green), and the first-order accurate method (blue). On the left, the solution is obtained on a coarse grid with a small Courant number, while on the right, it is computed on a fine mesh with a large Courant number.
• Figure 4: Comparisons for the example in Section \ref{['sec-eoc']}: the curves correspond to the exact solution (orange), the direct method in Remark \ref{['rem-comparison']} (red), the inverse method (green), and the first-order method (blue), computed for $I=256$ and the Courant number of 16. The left plot shows the solution with a fixed parameter (unlimited), while the right one the high-resolution (limited) form.
• Figure 5: Nonsmooth initial condition $q^0(x)$ that consists of 4 specific shapes.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3