Seminar in Numerical Analysis: Giacomo De Souza (EPFL)
Traditional explicit Runge--Kutta schemes, though computationally inexpensive, are inefficient for the integration of stiff ordinary differential equations due to stability issues. Conversely, implicit schemes are stable but can be overly expensive due to the solution of possibly large nonlinear systems with Newton-like methods, whose convergence is neither guaranteed for large time steps. Explicit stabilized schemes such as the Runge--Kutta--Chebyshev method (RKC) represent a viable compromise, as the width of their stability domain grows quadratically with respect to the number of function evaluations, thus presenting enhanced stability properties with a reasonable computational cost. These methods are particularly efficient for systems arising from the space discretization of parabolic partial differential equations (PDEs).
The efficiency of these methods deteriorates as the system becomes stiffer, even if stiffness is induced by only few degrees of freedom. In the framework of discretized parabolic PDEs, the number of function evaluations has to be chosen inversely proportional to the smallest element size in order to achieve stability, thus largely wasting computational resources on locally-refined meshes. We first tackle this issue by replacing the right hand side of the PDE with an averaged force, which is obtained by damping the high modes down using the dissipative effect of the equation itself and which is cheap to evaluate. Combining RKC methods with the averaged force we give rise to multirate RKC schemes, for which the number of expensive function evaluations is independent of the small elements' size.
The stability properties of our method are demonstrated on a model problem and numerical experiments confirm that the stability bottleneck caused by a few of fine mesh elements can be overcome without sacrificing accuracy.
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