# On long time integration of the heat equation

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## Abstract

We construct space-time Petrov–Galerkin discretizations of the heat equation on an unbounded temporal interval, either right-unbounded or left-unbounded. The discrete trial and test spaces are defined using Laguerre polynomials in time and are shown to satisfy the discrete inf-sup condition. Numerical examples are provided.

## Keywords

Heat equation Long-time Laguerre polynomials Stability## Mathematics Subject Classification

35K05 65M60 65M12## Notes

### Acknowledgments

The author thanks Ch. Schwab for motivating the topic, and the anonymous referees for their helpful suggestions.

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