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This paper establishes and analyzes a second-order accurate numerical schemefor the nonlinear partial integrodifferential equation with a weakly singularkernel. In the time direction, we apply the Crank-Nicolson method for the timederivative, and the product-integration (PI) rule is employed to deal withRiemann-Liouville fractional integral. From which, the non-uniform meshes areutilized 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 toformulate a fully discrete implicit difference scheme, we employ a standardcentered difference formula for the second-order spatial derivative, and theGalerkin method based on piecewise linear test functions is used to approximatethe nonlinear convection term. Then we derive the existence and uniqueness ofnumerical 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, weutilize a fixed point iterative algorithm to calculate the discrete scheme.Finally, numerical experiments illustrate the feasibility and efficiency of theproposed scheme, in which numerical results are consistent with our theoreticalanalysis.

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