A scale-dependent finite difference approximation for time fractional differential equation

Abstract

This study proposes a scale-dependent finite difference method (S-FDM) to approximate the time fractional differential equations (FDEs), using Hausdroff metric to conveniently link the order of the time fractional derivative (α) and the non-uniform time intervals. The S-FDM is unconditional stable and exhibits a convergence rate on the order of 2-α. Numerical tests show that the S-FDM is superior to the standard methods with either uniform or non-uniform time steps in computing time or cost, accuracy, and convergence rate, especially for a large time range. Hence, although many numerical schemes have been developed in the last decades for various FDEs, the unique S-FDM proposed in this study fits the requirement of calculating anomalous transport in natural systems involving a large spatiotemporal scale, which might be the future direction to extend the application of FDEs especially in Earth sciences, the ideal testbed for FDEs.

Appendix: Formulation of numerical schemes to solve the fractional relaxation and diffusion equations

Anomalous relaxation and diffusion is commonly observed in field experiments, especially in unsaturated flow and transport processes. To describe the memory effect imbedded in anomalous diffusion, the Caputo time fractional relaxation equation is given by

$$ \left\{ {\begin{array}{ll} &{{}_{0}^{C} D_{t}^{\gamma } u(t) + Bu(t) = f(t), \, 0 \le n - 1 < \gamma \le n \le 2} \\ &{u^{\left( k \right)} (0) = u_{0}^{\left( k \right)} \, \left( {k = 0,1, \ldots ,n - 1} \right)} \\ \end{array} } \right. $$

(19)

where $0 < \gamma \le 1$ represents the order of the fractional derivative in time.

Similarly, the time fractional diffusion equation can be expressed by:

$$ \left\{ {\begin{array}{lll} &{\frac{{\partial^{\alpha } u(x,t)}}{{\partial t^{\alpha } }} = D(x,t)\frac{{\partial^{2} u(x,t)}}{{\partial x^{2} }} + f(x,t)} \\ &u(x,t) = h(x,t),x \in \partial \Omega , \, t > 0, \, \\ &u(x,0) = w(x,t), \, t = 0. \hfill \end{array} } \right. $$

(20)

where α is the fractional order and $\alpha \in (0,1]$, $x \in [0,L], \, t > 0$, D(x,t) [L²T^−α] is the diffusion coefficient, and f(x,t) represents the source/sink term.

It is known [49, 50] that the initial-value problem (19) can be converted to the following Volterra integral equation:

$$ u(t) = \sum\limits_{i = 0}^{n - 1} {u_{0}^{\left( j \right)} } \frac{{t^{j} }}{j!} + \frac{1}{\Gamma (\gamma )}\int\limits_{0}^{t} {\left( {t - \tau } \right)^{\gamma - 1} (f(\tau ) - Bu(\tau ))d} \tau $$

(21)

1.1 S-FDM

Here, denoting $\Delta_{t} u(x,t_{k} ) = u(x,t_{k + 1} ) - u(x,t_{k} ),{\text{ and }}b_{k,j} = k^{1/\alpha } - j^{1/\alpha }$, we obtain a numerical scheme of the rectangular quadrature formula at the power-law time

$$ \int\limits_{0}^{{t_{k + 1} }} {\left( {t_{k + 1} - \tau } \right)^{\gamma - 1} (f(\tau ) - Bu(\tau ))d} \tau \approx \sum\limits_{j = 0}^{k} {m_{j,j + 1} } (f(t_{j} ) - Bu_{j} ) $$

(22)

where $m_{j,k} = \frac{{\Delta t_{\gamma } }}{\gamma }(b_{n + 1,j}^{\gamma } - b_{n + 1,k + 1}^{\gamma } )$, and we can have a lower triangular matrix M_(n+1)*(n+2) as follows

$$ {\mathbf{M}} = \left[ {\begin{array}{*{20}c} {b_{1,0} } & {b_{1,1} } & 0 & \cdots & 0 & 0 \\ {b_{2,0} } & {b_{2,1} } & {b_{2,2} } & \cdots & 0 & 0 \\ \vdots & {} & \ddots & \ddots & {} & \vdots \\ {b_{n,0} } & {b_{n,1} } & {b_{n,2} } & \cdots & {b_{n,n} } & 0 \\ {b_{n + 1,0} } & {b_{n + 1,1} } & {b_{n + 1,2} } & \cdots & {b_{n + 1,n} } & {b_{n + 1,n + 1} } \\ \end{array} } \right]_{(n + 1) \times (n + 2)} $$

Here, we can also solve the relaxation equation through the implicit scheme after applying the scale-dependent difference for the Caputo time fractional derivative:

$$ \begin{aligned} &\frac{{\partial^{\gamma } u(t_{n + 1} )}}{{\partial t^{\gamma } }} \approx \frac{1}{\Gamma (1 - \gamma )}\sum\limits_{k = 0}^{n} {\frac{{\Delta_{t} u(t_{k} )}}{{\tau_{k + 1} }}} \int_{{t_{k} }}^{{t_{k + 1} }} {\frac{d\xi }{{(t_{n + 1} - \xi )^{\gamma } }}} \hfill \\ &\quad = \frac{{ - \Delta t_{\gamma }^{ - 1} }}{\Gamma (2 - \alpha )}\sum\limits_{k = 0}^{n} {\frac{{\Delta_{t} u(t_{k} )}}{{b_{k + 1,k} }}} [(b_{n + 1,k + 1} )^{1 - \gamma } - (b_{n + 1,k} )^{1 - \gamma } ] \hfill \\ \end{aligned} $$

(23)

where $\tau_{k} = t_{k} - t_{k - 1}$. The function u(t) in (23) is the solution of Eq. (1).

Combining the implicit finite difference scheme and Eq. (8), the time fractional relaxation equation with the implicit scheme can possess the numerical results with the S-FDM as follows:

n = 0

$$ \left( {\frac{{\Delta t_{\alpha }^{ - 1} }}{\Gamma (2 - \alpha )} + B} \right)u(t_{1} ) = \frac{{\Delta t_{\alpha }^{ - 1} }}{\Gamma (2 - \alpha )}u(t_{0} ) + f(t_{1} ) $$

n = 1

$$ \begin{aligned} & \left( {\frac{{\Delta t_{\alpha }^{ - 1} }}{\Gamma (2 - \alpha )} + B} \right)u(t_{2} ) = f(t_{2} ) \\ & \quad + \frac{{\Delta t_{\alpha }^{ - 1} }}{\Gamma (2 - \alpha )}\left( {\left( {\frac{{b_{2,1}^{1 - \alpha } }}{{b_{2,1} }} + b_{2,1}^{1 - \alpha } - b_{2,0}^{1 - \alpha } } \right)u(t_{1} ) - \left( {b_{2,1}^{1 - \alpha } - b_{2,0}^{1 - \alpha } } \right)u(t_{0} )} \right) \\ \end{aligned} $$

n≥2

$$ \begin{aligned} & \left( {\frac{{\Delta t_{\alpha }^{ - 1} }}{\Gamma (2 - \alpha )}\frac{{b_{n + 1,n}^{1 - \alpha } }}{{b_{n + 1,n} }} + B} \right)u(t_{n + 1} ) \\ & \quad = \frac{{\Delta t_{\alpha }^{ - 1} }}{\Gamma (2 - \alpha )}\left( \begin{aligned} \left( {\frac{{b_{n + 1,n}^{1 - \alpha } }}{{b_{n + 1,n} }} + \frac{{b_{n + 1,n}^{1 - \alpha } - b_{n + 1,n - 1}^{1 - \alpha } }}{{b_{n,n - 1} }}} \right)u(t_{n} ) \hfill \\ - \left( {b_{n + 1,1}^{1 - \alpha } - b_{n + 1,0}^{1 - \alpha } } \right)u(t_{0} ) + \sum\limits_{k = 1}^{n - 1} {u(t_{k} )} \hfill \\ \left( { - \frac{{b_{n + 1,k + 1}^{1 - \alpha } - b_{n + 1,k}^{1 - \alpha } }}{{b_{k + 1,k} }} + \frac{{b_{n + 1,k}^{1 - \alpha } - b_{n + 1,k - 1}^{1 - \alpha } }}{{b_{k,k - 1} }}} \right) \hfill \\ \end{aligned} \right) + f(t_{n + 1} ) \\ \end{aligned} $$

where t_i stands for the time position.

Compared with the numerical scheme at the clock time, the S-FDM appears to be complex and highly computationally efficient due to its complex formula containing the part of b_k,j. However, the complex formulas show interesting generalizations to accelerate the computational efficiency by reading a lower triangular matrix M_(n+1)*(n+1) and matrix N_(n+1)*(n+1) = M.^(1−α) and reach high precision in solutions:

$$ {\mathbf{M}} = \left[ {\begin{array}{*{20}c} {b_{1,0} } & 0 & \cdots & 0 & 0 \\ {b_{2,0} } & {b_{2,1} } & \cdots & 0 & 0 \\ \vdots & {} & \ddots & {} & \vdots \\ {b_{n,0} } & {b_{n,1} } & \cdots & {b_{n,n - 1} } & 0 \\ {b_{n + 1,0} } & {b_{n + 1,1} } & \cdots & {b_{n + 1,n - 1} } & {b_{n + 1,n} } \\ \end{array} } \right]_{{\left( {n + 1} \right) \times \left( {n + 1} \right)}} $$

The time fractional diffusion equation is:

n = 0

$$ \begin{aligned} & \left( {\frac{{\Delta t_{\alpha }^{ - 1} }}{\Gamma (2 - \alpha )} + \frac{2}{{\Delta x^{2} }}} \right)u(x_{i} ,t_{1} ) \\ & \quad - D\frac{{u(x_{i + 1} ,t_{1} ) + u(x_{i - 1} ,t_{1} )}}{{\Delta x^{2} }} = \frac{{\Delta t_{\alpha }^{ - 1} }}{\Gamma (2 - \alpha )}u(x_{i} ,t_{0} ) + f(x_{i} ,t_{1} ) \\ \end{aligned} $$

n = 1

$$ \begin{aligned} & \left( {\frac{{\Delta t_{\alpha }^{ - 1} }}{\Gamma (2 - \alpha )} + \frac{2}{{\Delta x^{2} }}} \right)u(x_{i} ,t_{2} ) \\ & \quad - D\frac{{u(x_{i + 1} ,t_{2} ) + u(x_{i - 1} ,t_{2} )}}{{\Delta x^{2} }} = f(x_{i} ,t_{2} ) \\ & \quad + \frac{{\Delta t_{\alpha }^{ - 1} }}{\Gamma (2 - \alpha )}\left( {\left( {\frac{{b_{2,1}^{1 - \alpha } }}{{b_{2,1} }} + b_{2,1}^{1 - \alpha } - b_{2,0}^{1 - \alpha } } \right)u(x_{i} ,t_{1} )} \right. \\ & \quad \left. { -\, \left( {b_{2,1}^{1 - \alpha } - b_{2,0}^{1 - \alpha } } \right)u(x_{i} ,t_{0} )} \right) \\ \end{aligned} $$

n≥2

$$ \begin{aligned} & \left( {\frac{{\Delta t_{\alpha }^{ - 1} }}{\Gamma (2 - \alpha )}\frac{{b_{n + 1,n}^{1 - \alpha } }}{{b_{n + 1,n} }} + \frac{2}{{\Delta x^{2} }}} \right)u(x_{i} ,t_{n + 1} ) \\ & \quad - D\frac{{u(x_{i + 1} ,t_{n + 1} ) + u(x_{i - 1} ,t_{n + 1} )}}{{\Delta x^{2} }} \\ & \quad = \frac{{\Delta t_{\alpha }^{ - 1} }}{\Gamma (2 - \alpha )}\left( \begin{aligned} \left( {\frac{{b_{n + 1,n}^{1 - \alpha } }}{{b_{n + 1,n} }} + \frac{{b_{n + 1,n}^{1 - \alpha } - b_{n + 1,n - 1}^{1 - \alpha } }}{{b_{n,n - 1} }}} \right)u(x_{i} ,t_{n} ) \hfill \\ - \left( {b_{n + 1,1}^{1 - \alpha } - b_{n + 1,0}^{1 - \alpha } } \right)u(x_{i} ,t_{0} ) + \sum\limits_{k = 1}^{n - 1} {u(x_{i} ,t_{k} )} \hfill \\ \left( { - \frac{{b_{n + 1,k + 1}^{1 - \alpha } - b_{n + 1,k}^{1 - \alpha } }}{{b_{k + 1,k} }} + \frac{{b_{n + 1,k}^{1 - \alpha } - b_{n + 1,k - 1}^{1 - \alpha } }}{{b_{k,k - 1} }}} \right) \hfill \\ \end{aligned} \right) + f(x_{i} ,t_{n + 1} ) \\ \end{aligned} $$

where x_i represents the space position, and t_i stands for the time position.

1.2 Uniform grid

Here, denoting $\Delta_{t} u(x,t_{k} ) = u(x,t_{k + 1} ) - u(x,t_{k} ),\; {\text{ and }}\; \tau = T /N, \, t_{k} = k\tau$, we obtain a numerical scheme for the Caputo derivative definition at the clock time

$$ \begin{aligned} & {}_{0}^{C} D_{t}^{\alpha } u(x,t_{k + 1} ) \approx \frac{1}{\Gamma (1 - \alpha )}\sum\limits_{j = 0}^{k} {\frac{{\Delta_{t} u(x,t_{j + 1} )}}{\tau }} \int_{j\tau }^{{\left( {j + 1} \right)\tau }} {\frac{d\xi }{{\left( {t_{k + 1} - \xi } \right)^{\alpha } }}} \\ & \quad = \frac{{\tau^{ - \alpha } }}{\Gamma (2 - \alpha )}\sum\limits_{j = 0}^{k} {\Delta_{t} u(x,t_{j + 1} )} \left[ {\left( {j + 1} \right)^{1 - \alpha } - j^{1 - \alpha } } \right], \\ \end{aligned} $$

Combining the implicit finite difference scheme and Eq. (8), the time fractional relaxation equation with the implicit scheme can possess the numerical results with the uniform grid as follows:

n = 0

$$ \begin{aligned} & \left( {\frac{{\tau^{ - \alpha } }}{\Gamma (2 - \alpha )} + \frac{2}{{\Delta x^{2} }}} \right)u(x_{i} ,t_{1} ) \\ & \quad - D\frac{{u(x_{i + 1} ,t_{1} ) + u(x_{i - 1} ,t_{1} )}}{{\Delta x^{2} }} = \frac{{\tau^{ - \alpha } }}{\Gamma (2 - \alpha )}u(x_{i} ,t_{0} ) + f(x_{i} ,t_{1} ) \\ \end{aligned} $$

n = 1

$$ \begin{aligned} & \left( {\frac{{\tau^{ - \alpha } }}{\Gamma (2 - \alpha )} + \frac{2}{{\Delta x^{2} }}} \right)u(x_{i} ,t_{2} ) - D\frac{{u(x_{i + 1} ,t_{2} ) + u(x_{i - 1} ,t_{2} )}}{{\Delta x^{2} }} = f(x_{i} ,t_{2} ) \\ & \quad + \frac{{{ - }\tau^{ - \alpha } }}{\Gamma (2 - \alpha )}\left( {\left( {2^{1 - \alpha } - 2} \right)u(x_{i} ,t_{1} ) - \left( {2^{1 - \alpha } - 1} \right)u(x_{i} ,t_{0} )} \right) \\ \end{aligned} $$

n>2

$$ \begin{aligned} & \left( {\frac{{\tau^{1 - \alpha } }}{\Gamma (2 - \alpha )} + \frac{2}{{\Delta x^{2} }}} \right)u(x_{i} ,t_{n + 1} ) - D\frac{{u(x_{i + 1} ,t_{n + 1} ) + u(x_{i - 1} ,t_{n + 1} )}}{{\Delta x^{2} }} \\ & \quad = \frac{{ - \tau^{1 - \alpha } }}{\Gamma (2 - \alpha )}\left( \begin{aligned} \left( {2^{1 - \alpha } - 2} \right)u(x_{i} ,t_{n} ) \hfill \\ - \left( {\left( {n + 1} \right)^{1 - \alpha } - n^{1 - \alpha } } \right)u(x_{i} ,t_{0} ) + \sum\limits_{k = 1}^{n - 1} {u(x_{i} ,t_{n - k} )} \hfill \\ \left( {\left( {k + 2} \right)^{1 - \alpha } - 2\left( {k + 1} \right)^{1 - \alpha } + k^{1 - \alpha } } \right) \hfill \\ \end{aligned} \right) + f(x_{i} ,t_{n + 1} ) \\ \end{aligned} $$

where x_i represents the space position, and t_i stands for the time.