Asymptotic Solution of Stiff PDEs with the CSP Method: The Reaction Diffusion Equation
SIAM Journal on Scientific Computing
Journal of Computational Physics
Journal of Computational Physics
The G-Scheme: A framework for multi-scale adaptive model reduction
Journal of Computational Physics
Eigenvalues of the Jacobian of a Galerkin-Projected Uncertain ODE System
SIAM Journal on Scientific Computing
Journal of Computational Physics
Multiscale Stochastic Preconditioners in Non-intrusive Spectral Projection
Journal of Scientific Computing
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In systems of stiff Ordinary Differential Equations (ODEs) both fast and slow time scales are encountered. The fast time scales are responsible for the development of low-dimensional manifolds on which the solution moves according to the slow time scales. In this paper, methodologies for constructing highly accurate (i) expressions describing the manifold, and (ii) simplified non-stiff equations governing the slow evolution of the solution on the manifold are developed, according to an iterative procedure proposed in the Computational Singular Perturbation (CSP) method. It is shown that the increasing accuracy achieved with each iteration is directly related to the time rates of change of the CSP vectors spanning the manifold along the solution trajectory. Here, an algorithm is presented which implements these calculations and is validated on the basis of two simple examples.