Computational frameworks for the fast Fourier transform
Computational frameworks for the fast Fourier transform
Time-Marching Algorithms for Nonlocal Evolution Equations Based Upon "Approximate Approximations"
SIAM Journal on Scientific Computing
Pseudospectra of Linear Operators
SIAM Review
SIAM Journal on Applied Mathematics
Stability of pole solutions for planar propagating flames: I. exact eigenvalues and eigenfunctions
SIAM Journal on Applied Mathematics
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
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The aim of this work is the accurate calculation of periodic solutions to the Sivashinsky equation, which models dynamics of the long wave instability of laminar premixed flame. A highly accurate computational algorithm was developed in both one and two spatial dimensions and its crucial implementation details are presented. The algorithm is based on the concept of saturated asymptotic approximations and can be straightforwardly extended to a wide variety of nonlinear integro-differential equations. The development of such an algorithm was motivated by difficulties in interpretation of the results of numerical experiments with the Sivashinsky equation using spectral methods. The computations carried out by the algorithm in question are in good agreement with the results obtained earlier by spectral methods. Analysis of the accuracy of obtained numerical solutions and of their stabilization to steady states supports the idea of the instability of the steady coalescent pole solutions (with maximal possible number of poles) to the Sivashinsky equation in large domains through huge linear transient amplification of nonmodal perturbations of small but finite amplitudes.