A higher-order uniform accuracy scheme for nonlinear $\psi$-Volterra integral equations in two dimension with weakly singular kernel

In this paper, we proposed a higher-order uniform accuracy scheme for nonlinear $\psi$-Volterra integral equations in two dimension with weakly singular kernel by using the modified block-by-block method. First, we constructed a high order uniform accuracy scheme method in this paper by dividing the entire domain into some small sub-domains and approximating the integration function with biquadratic interpolation in each sub-domain. Second, we rigorously proved that the convergence order of the higher order uniform accuracy scheme was $O(h_s^{3+{\sigma }_1}+h_t^{3+{\sigma }_2})$ with $0<{\sigma }_1,{\sigma }_2<1$ by using the discrete Gronwall inequality. Finally, two numerical examples were used to illustrate experimental results with different values of $\psi$ to support the theoretical results.