A combined Lax-Wendroff/interpolation approach with finite element method for a three-dimensional system of tectonic deformation model: application to landslides in Cameroon

TL;DR

This work addresses modeling landslide response to tectonic deformation via a three-dimensional elastodynamic system on a bounded domain $\Omega \subset \mathbb{R}^3$ with displacement $w$ and stress $\kappa$. It introduces a novel numerical method that couples a modified Lax-Wendroff/interpolation time discretization with a finite element spatial discretization, yielding updates $(28)$–$(31)$. Theoretical results establish stability under a CFL-type condition $k/h \le C_{sr}$ and convergence with temporal order two and spatial convergence $O(h^{p})$ (with $p \ge 2$). Numerical experiments, including landslide cases in Cameroon (Dschang cliff, Mbankolo, Gouache), validate the method and demonstrate its utility for hazard delineation and rapid risk assessment.

Abstract

This paper develops an efficient computational technique to assess the landslide responses to tectonic deformation and to predict the implications of large bedrocks landslides on the short and long-term development of the disasters. The considered equations represent a three-dimensional system of geological structure deformation subject to suitable initial and boundary conditions. The space derivatives are approximated using the finite element procedure while the approximation in time derivative is obtained using the Lax-Wendroff and interpolation techniques. The new approach is so called a combined Lax-Wendroff/interpolation method with finite element method. The modified Lax-Wendroff/interpolation scheme is employed to efficiently treat the time derivative term and to provide a suitable time step restriction for stability. Under this time step requirement, both stability and error estimates of the new algorithm are deeply analyzed using a constructed strong norm. The theory suggests that the developed computational technique is second-order accurate in time and spatial convergent with order O(h^{p}), where $h$ denotes the space size and p is a positive integer. A wide set of numerical examples are carried out to confirm the theoretical results and to demonstrate the utility and validity of the proposed numerical scheme. An application to landslides observed in west and center regions in Cameroon from October 2019 to November 2024, are discussed.

Figures (4)

• Figure 1: Geological structure deformation: Dschang cliff, Mbankolo and Gouache landslides in Cameroon.
• Figure 2: Graphs of error $(e_{h})$, displacement $(w_{h})$ and stress tensor $(\kappa_{h})$ corresponding to Example 1 (first four figures) and Example 2 (last four figures).
• Figure 3: Graphs of displacement and stress tensor related to Dschang cliff landslides: first four figures for $g_{c}=7.6\%$ and last four figures for $g_{c}=0.7km$.
• Figure 4: Graphs of displacement and stress tensor corresponding to Mbankolo landslides (first four figures) and Gouache landslides (last four figures).

Theorems & Definitions (11)

• Remark 2.1
• Remark 2.2
• Lemma 2.1
• proof
• Remark 2.3
• Lemma 2.2
• proof
• Theorem 3.1
• proof
• Theorem 3.2