Rigorous numerical computations for 1D advection equations with variable coefficients
It provides a foundational approach for verified numerical computations of hyperbolic PDEs, addressing a gap in rigorous computing for this class of problems.
This paper presents a verified computing methodology for 1D advection equations with variable coefficients, achieving high accuracy and speed in rigorous error estimates for hyperbolic PDEs.
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.