Rigorous numerical computations for 1D advection equations with variable coefficients

TL;DR

This paper introduces a verified numerical method for solving 1D advection equations with variable coefficients, combining spectral methods and semigroup theory to ensure high-accuracy solutions with rigorous error bounds.

Contribution

It presents a novel verified computational approach for hyperbolic PDEs using spectral methods and semigroup theory, filling a gap in rigorous numerical analysis for such equations.

Findings

High-accuracy error estimates demonstrated

Efficient application of semigroup theory in Fourier space

Validated numerical examples confirming method effectiveness

Abstract

This paper provides a methodology of verified computing for solutions to 1-dimensional advection equations with variable coefficients. The advection equation is typical partial differential equations (PDEs) of hyperbolic type. There are few results of verified numerical computations to initial-boundary value problems of hyperbolic PDEs. Our methodology is based on the spectral method and semigroup theory. The provided method in this paper is regarded as an efficient application of semigroup theory in a sequence space associated with the Fourier series of unknown functions. This is a foundational approach of verified numerical computations for hyperbolic PDEs. Numerical examples show that the rigorous error estimate showing the well-posedness of the exact solution is given with high accuracy and high speed.