Numerical approximations of first kind Volterra convolution equations with discontinuous kernels

Davies, Penny J. and Duncan, Dugald B. (2017) Numerical approximations of first kind Volterra convolution equations with discontinuous kernels. Journal of Integral Equations and Applications, 29 (1). pp. 41-73. ISSN 0897-3962 (https://doi.org/10.1216/JIE-2017-29-1-41)

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Abstract

The cubic "convolution spline'" method for first kind Volterra convolution integral equations was introduced in [Convolution spline approximations of Volterra integral equations, J. Integral Equations Appl., 26:369--410, 2014]. Here we analyse its stability and convergence for a broad class of piecewise smooth kernel functions and show it is stable and fourth order accurate even when the kernel function is discontinuous. Key tools include a new discrete Gronwall inequality which provides a stability bound when there are jumps in the kernel function, and a new error bound obtained from a particular B-spline quasi-interpolant.

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