On the Computation of Blow-Up Solutions for Nonlinear Volterra Integrodifferential Equations

We make use of an adaptive numerical method to compute blow-up solutions for nonlinear ordinary Volterra integrodifferential equations (VIDEs). The method is based on the implicit midpoint method and the implicit Euler method and is named the implicit midpoint-implicit Euler (IMIE) method and was used to compute blow-up solutions in semilinear ODEs and parabolic PDEs in our earlier work. We demonstrate that the method produces superior results to the adaptive PECE-implicit Euler (PECE-IE) method and the MATLAB solver of comparable order just as it did in our previous contribution. We use quadrature rules to approximate the integral in the VIDE and demonstrate that the choice of quadrature rule has a significant effect on the blow-up time computed. In cases where the problem contains a convolution kernel with a singularity we use convolution quadrature.