A numerical time-stepping algorithm for differential or partial differential equations is proposed that adaptively modifies the dimensionality of the underlying modal basis expansion.Specifically,the method takes advantage of any underlying low-dimensional manifolds or subspaces in the system by using dimensionality-reduction techniques,such as the proper orthogonal decomposition,in order to adaptively represent the solution in the optimal basis modes.The method can provide significant computational savings for systems where low-dimensional manifolds are present since the reduction can lower the dimensionality of the underlying high-dimensional system by orders of magnitude.A comparison of the computational efficiency and error for this method are given showing the algorithm to be potentially of great value for high-dimensional dynamical systems simulations,especially where slow-manifold dynamics are known to arise.The method is envisioned to automatically take advantage of any potential computational saving associated with dimensionality-reduction,much as adaptive time-steppers automatically take advantage of large step sizes whenever possible.
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