A robust time-split linearized explicit/implicit technique for two-dimensional hydrodynamic model: an application to floods in Cameroon far north region

TL;DR

The paper addresses robustly simulating two-dimensional shallow water flows with moving boundaries by introducing a time-split linearized explicit/implicit scheme based on locally one-dimensional (LOD) splitting. The method forms a symmetric composition $\mathcal{P}(k)=\mathcal{P}_{1}(k/2)\mathcal{P}_{2}(k)\mathcal{P}_{1}(k/2)$, achieving spatial fourth-order accuracy and temporal second-order convergence, aided by high-order difference operators and nonlinear flux/source terms. A CFL-type time-step restriction is derived for stability, with $k \le \frac{48}{\gamma}\min\left\{\frac{\|\overline{\beta}\|_{0}}{\sqrt{M_{x}-3}\,\||\overline{u}|\|_{0,\infty}},\frac{\||\overline{u}|\|_{0,\infty}}{\||\overline{u}^{2}+\frac{1}{2}g\overline{h}|\|_{0,\infty}}\right\}\Delta x$ and $0<\gamma\le 18$, along with bounds on $\rho_{\max}(A)$. The middle implicit step is unconditionally stable, while the explicit steps require the time-step restriction. Numerical experiments, including floods in Cameroon, confirm second-order temporal and fourth-order spatial convergence under the stability condition and demonstrate the method’s practical utility for coastal-plain flood risk assessment and planning.

Abstract

This paper deals with a time-split explicit/implicit approach for solving a two-dimensional hydrodynamic flow model with appropriate initial and boundary conditions. The time-split technique is employed to upwind the convection term and to treat the friction slope so that the numerical oscillations and stability are well controlled. A suitable time step restriction for stability and convergence accurate of the new algorithm is established using the $L^{\infty}(0,T; L^{2})$-norm. Under a time step requirement, some numerical examples confirm the theoretical studies and suggest that the proposed computational technique is spatial fourth-order accurate and temporal second-order convergent. An application to floods observed in Cameroon far north region is considered and discussed.

Figures (5)

• Figure 1: Dam-break, Water depth and Geometric configuration.
• Figure 2: Situation of floods, Logone river and Guere lake in Mayo-Danay.
• Figure 3: Graphs of water depth, x- and y-direction velocities and errors corresponding to Example 1.
• Figure 4: Graphs of water depth, x- and y-direction velocities and errors corresponding to Example 2.
• Figure 5: Graphs of water depth, x- and y-direction velocities for floods in Logone river.

Theorems & Definitions (12)

• Remark 2.1
• Lemma 2.1
• proof
• Remark 2.2
• Theorem 3.1
• Remark 3.1
• Definition 3.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3