1,065 research outputs found
Algebraic construction of a third order difference approximations for fractional derivatives and applications
Finite difference approximations for fractional derivatives based on Grunwald formula are well known to be of first order accuracy, but display unstable solutions with known numerical methods. The shifted form of the Grunwald approximation removes this instability and keeps the same first order accuracy. Higher order approximations have been obtained by convex combinations of various shifted Grunwald approximations. Recently, a second order shifted Grunwald type approximation was constructed algebraically through a generating function. In this paper, we derive a new third order approximation from this second order approximation by preconditioning the fractional differential operator. This approximation is used with Crank-Nicolson numerical scheme to approximate the solutions of space-fractional diffusion equations by the same preconditioning. Stability and convergence of the numerical scheme are analysed, supported by numerical results showing third order convergence.
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Fourth order quasi-compact difference schemes for (tempered) space fractional diffusion equations
The continuous time random walk (CTRW) underlies many fundamental processes
in non-equilibrium statistical physics. When the jump length of CTRW obeys a
power-law distribution, its corresponding Fokker-Planck equation has space
fractional derivative, which characterizes L\'{e}vy flights. Sometimes the
infinite variance of L\'{e}vy flight discourages it as a physical approach;
exponentially tempering the power-law jump length of CTRW makes it more
`physical' and the tempered space fractional diffusion equation appears. This
paper provides the basic strategy of deriving the high order quasi-compact
discretizations for space fractional derivative and tempered space fractional
derivative. The fourth order quasi-compact discretization for space fractional
derivative is applied to solve space fractional diffusion equation and the
unconditional stability and convergence of the scheme are theoretically proved
and numerically verified. Furthermore, the tempered space fractional diffusion
equation is effectively solved by its counterpart of the fourth order
quasi-compact scheme; and the convergence orders are verified numerically.Comment: 27 pages, 1 figur
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