Rigorous numerical computations for 1D advection equations with variable coefficients
This paper provides a methodology of verified computing for solutions to 1D 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.
Keywords1D variable coefficient advection equation Verified numerical computation \(C_0\) semigroup Rigorous error bound Fourier–Chebyshev spectral method
Mathematics Subject Classification65G40 65M15 65M70 35L04
The authors express their sincere gratitude to Dr. M. Sobajima in Tyokyo University of Science for his essential suggestion on generating the \(C_0\) semigroup on sequence spaces. The authors would also like to express their gratitude to the anonymous referees for providing valuable comments that improve the paper. This work was partially supported by JSPS Grant-in-Aid for Early-Career Scientists, No. 18K13453.
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