Graded L1-2-3 time stepping and adaptive tension splines for two-dimensional time-fractional reaction–diffusion equations
Abstract
Abstract High-order schemes for two-dimensional time-fractionalreaction--diffusion equations are commonly built from an L-type approximation ofthe Caputo derivative on a uniform mesh together with a ''fourth-order tensionspline'' spatial operator. We revisit both ingredients: the tension parameterbecomes an active degree of freedom, chosen from the solution itself, and thetemporal mesh is graded.We prove that the parameter pair $(a,b)=(\frac1{12},\frac5{12})$, adoptedthroughout the non-polynomial spline literature to obtain fourth-order accuracy,is not attainable by any admissible tension parameter, real or imaginary: thespline parameters satisfy $a+b=\tan(\omega/2)/\omega$, which equals $\frac12$only in the polynomial limit, where $a=\frac16$. Thus the resulting operatorreduces to the classical nine-point compact scheme, in which the tension doesnot appear. We then derive the truncation error of the genuine tension-splineoperator on anisotropic grids and show that it is exact whenever the solutionprofile lies in the spline space. This yields an adaptive method in which thetension is recovered from local mode information and a computable indicatorselects, in each direction, the branch with the smaller predicted error.For time discretisation we construct the L1-2-3 formula on graded temporalmeshes, show that a direct implementation suffers from catastrophiccancellation, and give a stable evaluation. Numerical experiments forfractional orders $\alpha=0.2,0.5,0.8$ on isotropic…
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