A Rigorous Comparative Analysis of Convergence, Stability, and Entropy Decay for Finite Difference and Spectral Solvers of Nonlinear Volterra-Fredholm Integral Equations

Abstract

In addition to a detailed analysis of the Legendre-Galerkin method and finite difference scheme utilizing the composite trapezoidal rule, we consider some important aspects of the problem and prove their applicability using tools from approximation theory and functional analysis. In particular, we show that spectral methods are not only faster and better-conditioned but also dynamically simpler (that is, they have lower entropy) compared to other methods. Thus, we provide a justification for the choice of spectral methods whenever a smooth solution is expected.. Our aim is not to discuss implementation tricks, but rather to examine carefully how these two frameworks behave in terms of convergence speed, numerical stability, and the decay of topological entropy. The results show that for smooth (analytic) solutions, the spectral method converges exponentially, i.e.
, while the FD method is stuck with the much slower algebraic rate
2 . More importantly, the spectral discretisation enjoys a condition number that grows only as , whereas the FD system suffers from a quadratic growth
2 . On the entropy side, we prove that the spectral iteration map has exponentially decaying entropy (
num
num
), in sharp contrast to the algebraic decay observed for FD (
num

2 ). These findings provide a solid mathematical explanation for the well-known practical observation that spectral solvers become cheaper and more reliable as we increase the resolution, and they support the use of entropy as a meaningful indicator of solver performance