Fixed-Point Iteration for Nonlinear Volterra Equations in Computational Physics: A Strictly Pseudo-Contractive Mann Framework with Application to Heat Conduction with Memory
Nonlinear Volterra integral equations (VIEs) of the second kind arise in numerous physical contexts, including nonlinear heat conduction with memory, viscoelasticity, and population dynamics. However, when the kernel satisfies only a one-sided Lipschitz condition, classical contraction-based numerical methods fail, and efficient, provably convergent solvers are lacking. In this paper, we develop a fully discrete, computationally efficient numerical scheme for such equations. The method combines an implicit Euler time discretisation with an inexact Mann inner solver that requires no Jacobian evaluations, making it ideally suited for large-scale parallel computations. We establish global convergence of the adaptive, fully discrete scheme under realistic smoothness assumptions, and introduce a novel adaptive time-stepping strategy based on the Mann residual. The performance of the method is demonstrated on a physically motivated model of one-dimensional (1D) nonlinear heat conduction with memory. Extensive numerical comparisons against Picard iteration and Newton’s method show that the Mann-based solver is robust for large time steps, scales efficiently to high-resolution spatial discretisations, and maintains linear convergence rates as predicted by theory. Our open-source implementation provides a practical, ready-to-use tool for the computational physics community, and the framework is easily extended to partial integro-differential equations (PIDEs) and fractional-order memory kernels.