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Reuse of standard preconditioners for higher-order time discretizations of parabolic PDEs

In this work we study a preconditioned iterative method for some higher-order time discretizations of linear parabolic partial differential equations. We use the Padé approximations of the exponential function to discretize in time and show that standard solution algorithms for lower-order time discretization schemes, such as Crank–Nicolson and implicit Euler, can be reused as preconditioners for the arising linear system. The proposed preconditioner is order optimal with respect to the discretization parameters.