Spectral methods using rational basis functions on an infinite interval
Journal of Computational Physics
The Hermite spectral method for Gaussian-type functions
SIAM Journal on Scientific Computing
The Chebyshev-Legendre method: implementing Legendre methods on Chebyshev points
SIAM Journal on Numerical Analysis - Special issue: the articles in this issue are dedicated to Seymour V. Parter
A Rational Approximation and Its Applications to Differential Equations on the Half Line
Journal of Scientific Computing
Sparse grid spaces for the numerical solution of the electronic Schrödinger equation
Numerische Mathematik
Strang Splitting for the Time-Dependent Schrödinger Equation on Sparse Grids
SIAM Journal on Numerical Analysis
Algorithm 887: CHOLMOD, Supernodal Sparse Cholesky Factorization and Update/Downdate
ACM Transactions on Mathematical Software (TOMS)
Superfast Multifrontal Method for Large Structured Linear Systems of Equations
SIAM Journal on Matrix Analysis and Applications
Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Problems
SIAM Journal on Scientific Computing
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This is the second part in a series of papers on using spectral sparse grid methods for solving higher-dimensional PDEs. We extend the basic idea in the first part [J. Shen and H. Yu, SIAM J. Sci. Comp., 32 (2010), pp. 3228-3250] for solving PDEs in bounded higher-dimensional domains to unbounded higher-dimensional domains and apply the new method to solve the electronic Schrödinger equation. By using modified mapped Chebyshev functions as basis functions, we construct mapped Chebyshev sparse grid methods which enjoy the following properties: (i) the mapped Chebyshev approach enables us to build sparse grids with Smolyak's algorithms based on nested, spectrally accurate quadratures and allows us to build fast transforms between the values at the sparse grid points and the corresponding expansion coefficients; (ii) the mapped Chebyshev basis functions lead to identity mass matrices and very sparse stiffness matrices for problems with constant coefficients and allow us to construct a matrix-vector product algorithm with quasi-optimal computational cost even for problems with variable coefficients; and (iii) the resultant linear systems for elliptic equations with constant or variable coefficients can be solved efficiently by using a suitable iterative scheme. Ample numerical results are presented to demonstrate the efficiency and accuracy of the proposed algorithms.