Roundoff error problem in L2-type methods for time-fractional problems
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Roundoff error problems have occurred frequently in interpolation methods of time-fractional equations, which can lead to undesirable results such as the failure of optimal convergence. These problems are essentially caused by catastrophic cancellations. Currently, a feasible way to avoid these cancellations is using the Gauss–Kronrod quadrature to approximate the integral formulas of coefficients rather than computing the explicit formulas directly for example in the L2-type methods. This nevertheless increases computational cost and arises additional integration errors. In this work, a new framework to handle catastrophic cancellations is proposed, in particular, in the computation of the coefficients for standard and fast L2-type methods on general nonuniform meshes. We propose a concept of δ-cancellation and then some threshold conditions ensuring that δ-cancellations will not happen. If the threshold conditions are not satisfied, a Taylor-expansion technique is proposed to avoid δ-cancellation. Numerical experiments show that our proposed method performs as accurate as the Gauss–Kronrod quadrature method and meanwhile much more efficient. This enables us to complete long time simulations with hundreds of thousands of time steps in short time.
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