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.
Highlights
IntroductionThe spectral method has been used in many fields to solve linear partial differential equations (PDEs) such as the Poisson equation, the Helmholtz equation, and the diffusion equation [1,2]
The spectral method has been used in many fields to solve linear partial differential equations (PDEs) such as the Poisson equation, the Helmholtz equation, and the diffusion equation [1,2].The advantage of the spectral method over other numerical methods in solving linear Partial Differential Equation (PDE) is its high accuracy; when solutions of PDEs are smooth enough, errors of numerical solutions decrease exponentially as the number of discretization nodes increases [3].Based on the types of boundary conditions, different spectral basis functions can be used to discretize in physical space
We will use the quasi-inverse technique with the Chebyshev-Galerkin method to solve multi-dimensional problems for two reasons: (1) using the Galerkin basis function is much more convenient to apply the matrix diagonalization method because the Chebyshev-tau method needs to impose boundary conditions to the equation, which makes the entire calculation complicated in multi-dimensional problems and sometimes makes equations non-separable, and (2) the Chebyshev-Galerkin method is faster than the Chebyshev-tau method to obtain numerical solutions of PDEs
Summary
The spectral method has been used in many fields to solve linear partial differential equations (PDEs) such as the Poisson equation, the Helmholtz equation, and the diffusion equation [1,2]. As a result, when the Chebyshev-Galerkin method is used, tau lines are not created in the system of equation, and the interaction of tau lines in numerical calculations is not a concern Because of this advantage, the Chebyshev-Galerkin method is popular for spectral calculation of PDEs. For example, Shen [12] studied proper choices of Galerkin basis functions satisfying standard boundary conditions such as Dirichlet and Neumann boundary conditions and solved linear PDEs with them, and Julien and Watson [13] and Liu et al [11] used the Galerkin basis functions to solve multi-dimensional linear PDEs with high efficiency. We will explain the Chebyshev-tau method, the Chebyshev-Galerkin method, and the quasi-inverse technique by applying them to solve a 1-D Poisson equation
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