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An Efficient Spectral Method to Solve Multi-Dimensional Linear Partial Different Equations Using Chebyshev Polynomials

Author

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  • Sahuck Oh

    (School of Aerospace and Mechanical Engineering, Korea Aerospace University, Goyang 10540, Korea)

Abstract

We present a new method to efficiently solve a multi-dimensional linear Partial Differential Equation (PDE) called the quasi-inverse matrix diagonalization method. In the proposed method, the Chebyshev-Galerkin method is used to solve multi-dimensional PDEs spectrally. Efficient calculations are conducted by converting dense equations of systems sparse using the quasi-inverse technique and by separating coupled spectral modes using the matrix diagonalization method. When we applied the proposed method to 2-D and 3-D Poisson equations and coupled Helmholtz equations in 2-D and a Stokes problem in 3-D, the proposed method showed higher efficiency in all cases than other current methods such as the quasi-inverse method and the matrix diagonalization method in solving the multi-dimensional PDEs. Due to this efficiency of the proposed method, we believe it can be applied in various fields where multi-dimensional PDEs must be solved.

Suggested Citation

  • Sahuck Oh, 2019. "An Efficient Spectral Method to Solve Multi-Dimensional Linear Partial Different Equations Using Chebyshev Polynomials," Mathematics, MDPI, vol. 7(1), pages 1-21, January.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:1:p:90-:d:198218
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    Cited by:

    1. Musawenkhosi Patson Mkhatshwa & Melusi Khumalo, 2022. "Trivariate Spectral Collocation Approach for the Numerical Solution of Three-Dimensional Elliptic Partial Differential Equations," Mathematics, MDPI, vol. 10(13), pages 1-23, June.

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