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**Spectral methods in time for parabolic problems.**
*(English)*
Zbl 0668.65090

Spectral methods can provide a very useful tool for the solution of time- dependent partial differential equations. The standard scheme uses spectral methods to approximate the space derivatives and a finite difference approach to march the solution in time. These methods result in a unbalanced scheme, which has infinite accuracy in space and finite accuracy in time. Implicit algorithms based in this method involve inverting matrices. Using spectral methods for the space discretization results in full matrices, and inverting these matrices is a time- consuming procedure.

In the article an explicit scheme for solving a linear periodic parabolic system is described, when the space discretization is done by pseudospectral methods. The proposed scheme has an infinity accuracy both in time and space. This high accuracy and efficiency is achieved while the time resolution parameter N \((N=O(1/\Delta x)\) must satisfy \(M=O(N^{1+\epsilon})\) \(\epsilon >0\), compared to the common stability condition \(M=O(N^ 2)\), which must be satisfied in any explicit finite- order time algorithm.

In § 2 of the paper author presents a model problem and its fully discrete solution, in § 3 a new approach for approximating the evolution operator. In § 4 the error and stability analysis is given and in § 5 the numerical experiments conforming theoretical results are presented.

In the article an explicit scheme for solving a linear periodic parabolic system is described, when the space discretization is done by pseudospectral methods. The proposed scheme has an infinity accuracy both in time and space. This high accuracy and efficiency is achieved while the time resolution parameter N \((N=O(1/\Delta x)\) must satisfy \(M=O(N^{1+\epsilon})\) \(\epsilon >0\), compared to the common stability condition \(M=O(N^ 2)\), which must be satisfied in any explicit finite- order time algorithm.

In § 2 of the paper author presents a model problem and its fully discrete solution, in § 3 a new approach for approximating the evolution operator. In § 4 the error and stability analysis is given and in § 5 the numerical experiments conforming theoretical results are presented.

Reviewer: J.Vaníček

### MSC:

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

65N40 | Method of lines for boundary value problems involving PDEs |

35K20 | Initial-boundary value problems for second-order parabolic equations |

65N15 | Error bounds for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |