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A Crank-Nicolson-type compact difference method with the uniform time step for a class of weakly singular parabolic integro-differential equations
- Source :
- Applied Numerical Mathematics. 172:566-590
- Publication Year :
- 2022
- Publisher :
- Elsevier BV, 2022.
-
Abstract
- This paper is concerned with an efficient numerical method for a class of parabolic integro-differential equations with weakly singular kernels. Due to the presence of the weakly singular kernel, the exact solution has singularity near the initial time t = 0 . A generalized Crank-Nicolson-type scheme for the time discretization is proposed by designing a product integration rule for the integral term, and a compact difference approximation is used for the space discretization. The proposed method is constructed on the uniform time mesh, but it can still achieve the second-order convergence in time for weakly singular solutions. The unconditional stability and convergence of the method is proved and the optimal error estimate in the discrete L 2 -norm is obtained. The error estimate shows that the method has the second-order convergence in time and the fourth-order convergence in space. The extension of the method to two-dimensional problems is also discussed. A simple comparison is made with several existing methods. Numerical results confirm the theoretical analysis result and show the effectiveness of the proposed method.
Details
- ISSN :
- 01689274
- Volume :
- 172
- Database :
- OpenAIRE
- Journal :
- Applied Numerical Mathematics
- Accession number :
- edsair.doi...........b1b987f2fe47b0b9a12fa300bdf187cd