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An adaptive ROM approach for solving transfer equations

In this article, we present an adaptive method for solving transfer equations. The method consists in projecting the discretized problem on a basis we have defined in order to obtain a reduced model that can be quickly and accurately solved with classic numerical schemes. The originality of the methods stays in the way of the basis is constructed. At each iteration of computation, the basis is adapted: first the old basis is improved using a Karhunen-Loève decomposition whereas in a second phase the improved basis is expanded with Krylov vectors. The example we study is the one-dimension Burgers’ equation. The results we obtained were compared to the Newton-Raphson method: whereas the accuracy is not better than the Newton-Raphson method, we show that the computationnal time is drastically reduced. In addition, the basis we obtain shows a great ability to represent the long-time dynamics of the system, as shown in the last part of the paper.