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• W2342235926 abstract "This paper presents a modification of Krylov Subspace Spectral (KSS) Methods, which build on the work of Golub, Meurant and others per- taining to moments and Gaussian quadrature to pro- duce high-order accurate approximate solutions to the variable-coefficient second-order wave equation. Whereas KSS methods currently use Lanczos iter- ation to compute the needed quadrature rules, the modification uses block Lanczos iteration in order to avoid the need to compute two quadrature rules for each component of the solution, or use perturbations of quadrature rules that tend to be sensitive in prob- lems with oscillatory coefficients or data. It will be shown that under reasonable assumptions on the co- efficients of the problem, a 1-node KSS method is second-order accurate and unconditionally stable, and methods with more than one node are shown to pos- sess favorable stability properties as well, in addition to very high-order temporal accuracy. Numerical re- sults demonstrate that block KSS methods are signif- icantly more accurate than their non-block counter- parts, especially for problems that feature oscillatory coefficients. where p(x) is a positive function and q(x) is a nonnegative (but nonzero) smooth function. It follows that L is self- adjoint and positive definite. In (18, 20) a class of methods, called Krylov subspace spectral (KSS) methods, was introduced for the purpose of solving variable-coefficient parabolic problems. These methods are based on the application of techniques de- veloped by Golub and Meurant in (7), originally for the purpose of computing elements of the inverse of a matrix, to elements of the matrix exponential of an operator. It has been shown in these references that KSS methods, by employing different approximations of the solution oper- ator for each Fourier component of the solution, achieve higher-order accuracy in time than other Krylov subspace methods (see, for example, (14)) for stiff systems of ODE, and, as shown in (15), they are also quite stable, consid- ering that they are explicit methods. In (16), we considered whether these methods can be en- hanced, in terms of accuracy, stability or any other mea- sure, by using a single block Gaussian quadrature rule to compute each Fourier component of the solution, instead of two standard Gaussian rules. KSS methods take the solution from the previous time step into account only through a perturbation of initial vectors used in Lanczos iteration. While this enables KSS methods to handle stiff systems very effectively by giving individual attention to each Fourier component, and also yields high-order op- erator splittings (see (17)), it is worthwhile to consider whether it is best to use quadrature rules whose nodes are determined primarily by each basis function used to represent the solution, instead of the solution itself. Intu- itively, a block quadrature rule that uses a basis function and the solution should strike a better balance between the competing goals of computing each component with an approximation that is, in some sense, optimal for that component in order to deal with stiffness, and giving the solution a prominent role in computing the quadrature rules that are used to evolve it forward in time." @default.
• W2342235926 created "2016-06-24" @default.
• W2342235926 creator A5018418474 @default.
• W2342235926 date "2008-01-01" @default.
• W2342235926 modified "2023-09-27" @default.
• W2342235926 title "An Explicit, Stable, High-Order Spectral Method for the Wave Equation Based on Block Gaussian Quadrature" @default.
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