Computing Small Solutions of Burgers' Equation Backwards in Time.
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WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER
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The author constructs and analyzes an algorithm for the numerical computation of Burgers equation for preceding times, given an a-priori bound for the solution and an approximation to the terminal data. The method is based on the backward beam equation coupled with an iterative procedure for the solution of the non-linear problem via a sequence of linear problems. Also presented are the results of several numerical experiments. It turns out that the procedure converges asymptotically, i.e. in the same manner that an asymptotic expansion converges. This phenomenon seems related to the destruction of information, at t 0, which is typical in backwards dissipative equations. A-priori stability estimates are derived for the analytic backwards problem, and it is observed that in many numerical experiments, the distance backwards in time where significant accuracy can be attained is much larger than would be expected on the basis of such estimates. The method is useful for small solutions. Problems where steep gradients occur require considerably more precision in measurement. The algorithm is applicable to other semi-linear problems.
- Theoretical Mathematics