Abstract
The fractional KdV equation has significant physical implications, the study of its efficient method has great scientific significance. A fast compact difference (FCD) scheme on graded meshes is used to address the time-fractional nonlinear KdV equation with an initial singularity. The temporal discretization employs the fast L1 algorithm, while a fourth-order compact scheme is utilized for spatial discretization. The Newton's method is used to approximate the nonlinear term. The fast L1 algorithm can greatly shorten the CPU calculation time without losing the precision of numerical solution. We verify the existence and uniqueness of numerical solutions, while also provide unconditional stability and convergence analysis. Theoretical analysis is validated by numerical results, indicating the FCD scheme on graded meshes has high-accuracy and time-saving performance.
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