Numerical solution to Volterra integral equations of the first kind by implicit Runge-Kutta method of high accuracy

The analog of the implicit Runge-Kutta method applied to Volterra integral equations of the first kind is considered. It allows to obtain the results of high accuracy under a sufficient simplicity and stability of used algorithm. The estimation of numerical results for a fixed time step is performed. A special choice of integration's nodes and quadrature coefficients makes it possible to receive the error estimation, decreasing exponentially under increasing of a number of method's stages. That creates a good premise for using the method of high accuracy and permits to integrate with a big time step. The stability of method to variations of kernel and the right-hand side of the equation is proved. The above theoretical conclusions are confirmed by numerical experiments for Volterra equations of the first kind with different types of kernels and equivalent Volterra equation of the second kind.