7,053 research outputs found
Maximum norm error estimates of efficient difference schemes for second-order wave equations
AbstractThe three-level explicit scheme is efficient for numerical approximation of the second-order wave equations. By employing a fourth-order accurate scheme to approximate the solution at first time level, it is shown that the discrete solution is conditionally convergent in the maximum norm with the convergence order of two. Since the asymptotic expansion of the difference solution consists of odd powers of the mesh parameters (time step and spacings), an unusual Richardson extrapolation formula is needed in promoting the second-order solution to fourth-order accuracy. Extensions of our technique to the classical ADI scheme also yield the maximum norm error estimate of the discrete solution and its extrapolation. Numerical experiments are presented to support our theoretical results
The variable-step L1 scheme preserving a compatible energy law for time-fractional Allen-Cahn equation
In this work, we revisit the adaptive L1 time-stepping scheme for solving the
time-fractional Allen-Cahn equation in the Caputo's form. The L1 implicit
scheme is shown to preserve a variational energy dissipation law on arbitrary
nonuniform time meshes by using the recent discrete analysis tools, i.e., the
discrete orthogonal convolution kernels and discrete complementary convolution
kernels. Then the discrete embedding techniques and the fractional Gr\"onwall
inequality were applied to establish an norm error estimate on nonuniform
time meshes. An adaptive time-stepping strategy according to the dynamical
feature of the system is presented to capture the multi-scale behaviors and to
improve the computational performance.Comment: 17 pages, 20 figures, 2 table
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