What is it about?
A backward Euler alternating direction implicit (ADI) difference scheme is formulated and analyzed for the three-dimensional fractional evolution equation. In our method, the Riemann-Liouville fractional integral term is treated by means of first order convolution quadrature suggested by Lubich. Meanwhile, an ADI technique is adopted to reduce the multi-dimensional problem to a series of one-dimensional problems. A fully discrete difference scheme is constructed with space discretization by finite difference method. Two new inner products and corresponding norms are defined to analyze the scheme. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. Numerical experiments are reported to demonstrate the efficiency of our scheme.
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Why is it important?
Especially, we note that the three dimensional Peaceman-Reachford (P-R) scheme for parabolic equations is not unconditionally stable and only first order accurate in time \cite{Thomas}. In \cite{Douglas1}, Douglas and Rachford formulated an unconditionally stable scheme for three dimensional parabolic equations. Basing on the idea proposed in \cite{Douglas1}, we consider effective numerical methods for the three-dimensional fractional evolution equation.
Perspectives
In our method, The Riemann-Liouville fractional integral term is treated by means of first order convolution quadrature suggested by Lubich (see \cite{Lubich1}). The second order central difference scheme is applied for the spatial approximation. Meanwhile, an ADI technique is adopted to reduce the multi-dimensional problem to a series of one-dimensional problems. Two new inner products and corresponding norms are defined to analyze the scheme. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. The resulting finite difference scheme is unconditionally stable and convergent with the convergence order of 1 for time and 2 for space in new norms.
Hongbin Chen
Central South University of Forestry and Technology
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This page is a summary of: A backward Euler alternating direction implicit difference scheme for the three-dimensional fractional evolution equation, Numerical Methods for Partial Differential Equations, December 2017, Wiley,
DOI: 10.1002/num.22239.
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