  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.
