Abstract
A time-fractional initial-boundary value problem of wave type is considered, where the spatial domain is (0,1)d for some d∈{1,2,3} . Regularity of the solution 𝑢 is discussed in detail. Typical solutions have a weak singularity at the initial time t=0 : while 𝑢 and ut are continuous at t=0 , the second-order derivative utt blows up at t=0 . To solve the problem numerically, a finite difference scheme is used on a mesh that is graded in time and uniform in space with the same mesh size ℎ in each coordinate direction. This scheme is generated through order reduction: one rewrites the differential equation as a system of two equations using the new variable v:=ut ; then one uses a modified L1 scheme of Crank–Nicolson type for the driving equation. A fast variant of this finite difference scheme is also considered, using a sum-of-exponentials (SOE) approximation for the kernel function in the Caputo derivative. The stability and convergence of both difference schemes are analysed in detail. At each time level, the system of linear equations generated by the difference schemes is solved by a fast Poisson solver, thereby taking advantage of the fast difference scheme. Finally, numerical examples are presented to demonstrate the accuracy and efficiency of both numerical methods.
Award Identifier / Grant number: JBK2101012
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11671081
Award Identifier / Grant number: NSAF U1930402
Funding statement: The research of Jinye Shen is supported in part by the Fundamental Research Funds for the Central Universities (JBK2101012). The research of Zhi-Zhong Sun is supported in part by the National Natural Science Foundation of China under grant 11671081. The research of Martin Stynes is supported in part by the National Natural Science Foundation of China under grant NSAF U1930402.
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