Implementation of symmetrizers in ordinary differential equations

Noorhelyna Razali, Robert Peng Kong Chan


Two-step symmetrizers for the implicit midpoint and trapezoidal rules provide an alternative to the one-step smoothing formula for solving stiff ordinary differential equations. When used with the basic symmetric methods, these \(L\)-stable methods preserve the asymptotic error expansion in even powers of the step size and provide the necessary damping of oscillatory solutions. These new symmetrizers show effects similar to one-step smoothing but with the advantage of being order two. When generalized to higher order symmetric methods, such as the two-stage Gauss or the three-stage Lobatto IIIA, these symmetrizers can suppress order reduction for stiff problems. Here, we discuss one-step and two-step symmetrizers and their application in ordinary differential equations. We present numerical results with constant and variable step sizes that show the advantages of two-step symmetrizers over one-step symmetrizers of the implicit trapezoidal rule for stiff linear and nonlinear problems.

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Symmetrization; Implicit Trapezoidal Rule; Passive; Active

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