Numerical solution of linear Volterra integral equations of the second kind with sharp gradients

Collocation methods are a well developed approach for the numerical solution of smooth and weakly-singular Volterra integral equations. In this paper we extend these methods, through the use of partitioned quadrature based on the qualocation framework, to allow the efficient numerical solution of linear, scalar, Volterra integral equations of the second kind with smooth kernels containing sharp gradients. In this case the standard collocation methods may lose computational efficiency despite the smoothness of the kernel. We illustrate how the qualocation framework can allow one to focus computational effort where necessary through improved quadrature approximations, while keeping the solution approximation fixed. The computational performance improvement introduced by our new method is examined through several test examples. The final example we consider is the original problem that motivated this work: the problem of calculating the probability density associated with a continuous...

• Type: Journal
• Year: 2011
• Where: JCAM
• Authors: Samuel A. Isaacson, Robert M. Kirby
• Keywords: Collocation Methods, JCAM 2011, Probability Density, Volterra Integral Equations