Second-order accurate numerical scheme with graded meshes for the nonlinear partial integrodifferential equation arising from viscoelasticity

TL;DR

This paper develops a second-order accurate numerical scheme for a nonlinear partial integrodifferential equation from viscoelasticity, employing graded meshes, Crank-Nicolson, and Galerkin methods to handle singularities and nonlinearities effectively.

Contribution

It introduces a novel second-order scheme on graded meshes for viscoelasticity equations, with proven stability, convergence, and a fixed point iterative solver.

Findings

The scheme achieves second-order convergence in time.

Numerical results confirm the theoretical stability and accuracy.

The method effectively handles weakly singular kernels and nonlinear terms.

Abstract

This paper establishes and analyzes a second-order accurate numerical scheme for the nonlinear partial integrodifferential equation with a weakly singular kernel. In the time direction, we apply the Crank-Nicolson method for the time derivative, and the product-integration (PI) rule is employed to deal with Riemann-Liouville fractional integral. From which, the non-uniform meshes are utilized to compensate for the singular behavior of the exact solution at $t=0$ so that our method can reach second-order convergence for time. In order to formulate a fully discrete implicit difference scheme, we employ a standard centered difference formula for the second-order spatial derivative, and the Galerkin method based on piecewise linear test functions is used to approximate the nonlinear convection term. Then we derive the existence and uniqueness of numerical solutions for the proposed implicit difference scheme. Meanwhile, stability and convergence are proved by means of the discrete energy method. Furthermore, to demonstrate the effectiveness of the proposed method, we utilize a fixed point iterative algorithm to calculate the discrete scheme. Finally, numerical experiments illustrate the feasibility and efficiency of the proposed scheme, in which numerical results are consistent with our theoretical analysis.