In this section, we propose a novel finite volume scheme formulation (FVSF) for solving the SFCDE with constant coefficients

$$\begin{aligned} \frac{\partial\psi( x, t )}{\partial t} + \epsilon \frac{\partial^{\beta} \psi( x, t )}{\partial x^{\beta}} = \rho \frac{\partial^{2} \psi ( x, t )}{\partial x^{2}}, \end{aligned}$$

(21)

subject to the initial condition

$$\begin{aligned} \psi ( x,0 ) = g ( x ),\quad a \leq x \leq b, \end{aligned}$$

(22)

where \(t \geq0, 0< \beta\leq1, g ( x )\) is an analytical smooth function of spatial *x*, \(\psi ( x, t )\) is an unknown analytical function, *ϵ* and *ρ* are positive parameters, and *β* is a parameter that describes the order of the space fractional, where the space fractional derivative is described in the Riemann–Liouville sense. If \(\beta=1\), it becomes a linear convection–diffusion equation.

Now, to establish the finite volume scheme, we need to partition the finite domain \(\Omega=[ a, b ]\) to \(N +1\) uniformly spaced nodes \(x_{i} = a + ih, i =0,1,\dots, N\), where the spatial step is \(h = \frac{b - a}{N}\). Thus, by utilizing the Riemann–Liouville fractional derivative, we have

$$\begin{aligned} \frac{\partial\psi( x, t )}{\partial t} + \epsilon\frac{\partial }{\partial x} J_{a}^{1- \beta} \psi ( x, t ) = \rho \frac{\partial^{2} \psi ( x, t )}{\partial x^{2}},\quad 0< \beta \leq1, \end{aligned}$$

(23)

where \(J_{a}^{1- \beta}\) is the Riemann–Liouville integral with respect to *x*. Setting \(\alpha=1- \beta\) gives \(0\leq\alpha<1\). Integrating Eq. (23) over the *i*th control volume [\(x_{i - \frac{1}{2}}, x_{i + \frac{1}{2}} \)] suggests

$$\begin{aligned} \int_{x_{i - \frac{1}{2}}}^{x_{i + \frac{1}{2}}} \frac{\partial\psi( x, t )}{\partial t} \,dx= \int_{x_{i - \frac{1}{2}}}^{x_{i + \frac{1}{2}}} \frac{\partial}{\partial x} \biggl[- \epsilon J_{a}^{\alpha} \psi ( x, t ) + \rho\frac{\partial u ( x, t )}{\partial x} \biggr] \,dx. \end{aligned}$$

(24)

Dividing each side by *h* gives the standard finite volume discretization

$$\begin{aligned} \frac{d \overline{\psi}_{i} ( t )}{dt} = \frac{\epsilon}{h} \bigl[ J_{a}^{\alpha} \psi( x_{i - \frac{1}{2}}, t ) - J_{a}^{\alpha} \psi( x_{i + \frac{1}{2}}, t ) \bigr] + \frac{\rho}{h} \biggl[ \frac{\partial\psi ( x_{i + \frac{1}{2}}, t )}{\partial x} - \frac{\partial\psi ( x_{i - \frac{1}{2}}, t )}{\partial x} \biggr], \end{aligned}$$

(25)

where

$$\begin{aligned} \overline{\psi}_{i} ( t ) = \frac{1}{h} \int_{x_{i - \frac{1}{2}}}^{x_{i + \frac{1}{2}}} \psi ( x, t ) \,dx \end{aligned}$$

is the control volume averages of \(\psi( x, t )\).

Using the fractionally-shift Grünwald formula to approximate \(J_{a}^{\alpha} \psi( x, t )\) yields

$$\begin{aligned} J {}_{a}^{\alpha} \psi ( x, t ) = \lim h^{\alpha}_{ h \rightarrow 0} \sum_{j =0}^{ [ \frac{x - a}{h} + p ]} \frac{\Gamma( \alpha + j )}{\Gamma ( \alpha ) \Gamma( j +1)} \psi \bigl( x - ( j - p ) h, t \bigr), \end{aligned}$$

(26)

in which \(h = \frac{x - a}{N}, x \in ( a, b ] \). Hence, the Riemann–Liouville integral includes the order term

$$\begin{aligned} J_{a}^{\alpha} \psi ( x, t ) = h^{\alpha} \sum _{j =0}^{N + p} v_{j}^{\alpha} \psi \bigl( x - ( j - p ) h, t \bigr) + o ( 1 ). \end{aligned}$$

(27)

Therefore, due to certain needs of the fractional shift \(p = \frac{1}{2}\), we write

$$\begin{aligned} & J_{a}^{\alpha} \psi( x, t ) \vert_{x = x_{i - \frac{1}{2}}} = J_{a}^{\alpha} \psi ( x_{i - \frac{1}{2}}, t ) + h^{\alpha} \sum_{j =0}^{i} v_{j}^{\alpha} \psi ( x_{i - j}, t ) + o (1), \end{aligned}$$

(28)

$$\begin{aligned} & J_{a}^{\alpha} \psi( x, t ) \vert_{x = x_{i + \frac{1}{2}}} = J_{a}^{\alpha} \psi ( x_{i + \frac{1}{2}}, t ) + h^{\alpha} \sum_{j =0}^{i +1} v_{j}^{\alpha} \psi ( x_{i - j +1}, t ) + o (1), \end{aligned}$$

(29)

where \(v_{0}^{\alpha} =1, v_{1}^{\alpha} = \alpha\), and \(v_{j}^{\alpha} = ( 1- \frac{ ( 1- \alpha )}{j} ) v_{j -1}^{\alpha}, j =2,3,\dots\). For \(\frac{\partial\psi( x, t )}{\partial x}\), use the identity

$$\begin{aligned} \frac{\partial\psi ( x_{i}, t )}{\partial x} = \frac{\psi ( x_{i +1}, t ) - \psi ( x_{i -1}, t )}{2 h} +\mathcal{O} \bigl( h^{2} \bigr) \end{aligned}$$

(30)

to write

$$\begin{aligned} &\frac{\partial\psi ( x_{i - \frac{1}{2}}, t )}{\partial x} \approx\frac{\psi( x_{i + \frac{1}{2}}, t )- \psi( x_{i - \frac {3}{2}}, t )}{2 h}, \end{aligned}$$

(31)

$$\begin{aligned} &\frac{\partial\psi ( x_{i + \frac{1}{2}}, t )}{\partial x} \approx\frac{\psi ( x_{i + \frac{3}{2}}, t ) - \psi ( x_{i - \frac{1}{2}}, t )}{2 h}. \end{aligned}$$

(32)

For Eqs. (31) and (32), use the standard averaging scheme \(\psi ( x_{i + \frac{1}{2}}, t ) \approx\frac{[ \psi ( x_{i}, t ) + \psi ( x_{i +1}, t ) ]}{2}\) to construct approximations of first derivative in terms of function values at the nodes \(x_{j}\):

$$\begin{aligned} &\frac{\partial\psi ( x_{i - \frac{1}{2}}, t )}{\partial x} \approx\frac{1}{4 h} \bigl[ \psi ( x_{i}, t ) + \psi ( x_{i +1}, t ) - \psi ( x_{i -2}, t ) - \psi ( x_{i -1}, t ) \bigr], \end{aligned}$$

(33)

$$\begin{aligned} &\frac{\partial\psi ( x_{i + \frac{1}{2}}, t )}{\partial x} \approx\frac{1}{4 h} \bigl[ \psi ( x_{i +1}, t ) + \psi ( x_{i +2}, t ) - \psi ( x_{i -1}, t ) - \psi ( x_{i}, t ) \bigr]. \end{aligned}$$

(34)

So, Eq. (25) can be approximated as

$$\begin{aligned} \frac{d \overline{\psi}_{i} ( t )}{dt}={}&\epsilon h^{\alpha-1} \Biggl[ \sum _{j =0}^{i} v_{j}^{\alpha} \psi ( x_{i - j}, t ) - \sum_{j =0}^{i +1} v_{j}^{\alpha} \psi ( x_{i - j +1}, t ) \Biggr] \\ &{}+ \frac{\rho}{4 h^{2}} \bigl[ \psi ( x_{i -2}, t ) -2 \psi ( x_{i}, t ) + \psi( x_{i +2}, t ) \bigr]. \end{aligned}$$

(35)

Notice that if \(\psi( x, t )\) is a smooth function, then the value of the control volume averages \(\overline{\psi}_{i} ( t )\) will agree with the value of \(\psi( x, t )\) at the midpoint of the interval [\(x_{i - \frac{1}{2}}, x_{i + \frac{1}{2}} \)] to \(\mathcal{O} ( h^{2} ) \). So, we can rewrite Eq. (35) as

$$\begin{aligned} \frac{d\psi ( x_{i}, t )}{dt}={}&\epsilon h^{\alpha-1} \Biggl[ \sum _{j =0}^{i} v_{j}^{\alpha} \psi ( x_{i - j}, t ) - \sum_{j =0}^{i +1} v_{j}^{\alpha} \psi ( x_{i - j +1}, t ) \Biggr] \\ &{}+ \frac{\rho}{4 h^{2}} \bigl[ \psi ( x_{i -2}, t ) -2 \psi ( x_{i}, t ) + \psi ( x_{i +2}, t ) \bigr]. \end{aligned}$$

(36)

Now, define a temporal partition \(t_{n} = n\tau, n =0,1,\dots\), where *τ* is the time step, and use the standard backward difference to approximate the temporal derivative in Eq. (36) such that

$$\begin{aligned} \frac{d\psi ( x_{i}, t )}{dt} \bigg\vert _{t = t_{n +1}} = \frac{\psi ( x_{i}, t_{n +1} ) - \psi ( x_{i}, t_{n} )}{\tau} + \mathcal{O} ( \tau ). \end{aligned}$$

Let \(\psi_{i}^{n} \approx\psi( x_{i}, t_{n} )\) denote the numerical solution. Then we have

$$\begin{aligned} \frac{\psi_{i}^{n +1} - \psi_{i}^{n}}{\tau} ={}&\epsilon h^{\alpha-1} \Biggl[ \sum _{j =0}^{i} v_{j}^{\alpha} \psi_{i - j}^{n +1} - \sum_{j =0}^{i +1} v_{j}^{\alpha} \psi_{i - j +1}^{n +1} \Biggr] \\ &{}+ \frac{\rho}{4 h^{2}} \bigl[ \psi_{i -2}^{n +1} -2 \psi_{i}^{n +1} + \psi_{i +2}^{n +1} \bigr]. \end{aligned}$$

(37)

By collecting like terms, we rewrite Eq. (37) in the form

$$\begin{aligned} \frac{\psi_{i}^{n +1} - \psi_{i}^{n}}{\tau} = \frac{1}{h} \sum_{j =0}^{N} k_{ij} \psi_{j}^{n +1},\quad i =0,1,2,\dots, N, \end{aligned}$$

(38)

where \(k_{ij}\) is given by

$$\begin{aligned} k_{ij}= \textstyle\begin{cases} \epsilon h^{\alpha} [ v_{i - j}^{\alpha} - v_{i - j +1}^{\alpha} ],& j < i -2, \\ \epsilon h^{\alpha} [ v_{2}^{\alpha} - v_{3}^{\alpha} ] + \frac{\rho}{4 h},& j = i -2, \\ \epsilon h^{\alpha} [ v_{1}^{\alpha} - v_{2}^{\alpha} ],& j = i -1, \\ \epsilon h^{\alpha} [ v_{0}^{\alpha} - v_{1}^{\alpha} ] - \frac{\rho}{2 h},& j = i, \\ \epsilon h^{\alpha} [ - v_{0}^{\alpha} ],& j = i +1, \\ \frac{\rho}{4 h},& j = i +2, \\ 0,& j > i +2. \end{cases}\displaystyle \end{aligned}$$

The vector of the numerical solution \(\psi^{n} =[ \psi_{0}^{n}, \psi_{1}^{n},\dots, \psi_{N}^{n} ]\) can be denoted by

$$\begin{aligned} \biggl( I + \frac{\tau}{h} A \biggr) \psi^{n +1} = \psi^{n}, \end{aligned}$$

(39)

where the matrix *A* has the elements \(a_{ij} = - k_{ij}\).

The next two theorems explain that the finite volume scheme described in Eq. (37) is conditionally stable and consistent with first-order accuracy in time and second-order accuracy in space.

### Theorem 5

*For* \(i =0,1,\dots, N\), *the numerical scheme* \(\frac{\psi_{i}^{n +1} - \psi_{i}^{n}}{\tau} = \frac{1}{h} \sum_{j =0}^{N} k_{ij} \psi_{j}^{n +1}\) *is conditionally stable*.

### Proof

Substitute \(\psi_{i}^{n} = \hat{\psi}^{n} \exp ( iI\xi )\) into the numerical scheme to get

$$\begin{aligned} \frac{\psi_{i}^{n +1} - \psi_{i}^{n}}{\tau} = \frac{1}{h} \sum_{j =0}^{N} k_{ij} \psi_{j}^{n +1},\quad i =0,1,2,\dots, N. \end{aligned}$$

However, we have

$$\begin{aligned} \hat{\psi}^{n +1} \exp ( iI\xi ) - \psi^{n} \exp ( iI\xi ) = r \sum_{j =0}^{N} k_{ij} \hat{ \psi}^{n +1} \exp ( jI\xi ),\qquad r = \frac{\tau}{h}, \hat{ \psi}^{n +1} = \rho ( \xi ) \hat{\psi}^{n}, \end{aligned}$$

where

$$\begin{aligned} \rho ( \xi ) = \frac{1}{ [ 1- r \sum_{j =0}^{N} k_{ij} \exp ( ( j - i ) I\xi ) ]} \end{aligned}$$

satisfies the von Neumann condition when \(\vert 1- r \sum_{j =0}^{N} k_{ij} \exp ( ( j - i ) I\xi) \vert \geq1\). Therefore, by using the reverse triangle inequality, we have

$$\begin{aligned} \Biggl\vert 1- r \sum_{j =0}^{N} k_{ij} \exp\bigl( ( j - i ) I\xi\bigr) \Biggr\vert \geq\Biggl| 1- \Biggl| r \sum_{j =0}^{N} k_{ij} \exp\bigl( ( j - i ) I\xi\bigr) \Biggr| \Biggl| . \end{aligned}$$

Hence, the von Neumann condition will be satisfied whenever

$$\begin{aligned} \Biggl\vert 1-\Biggl\vert r \sum_{j =0}^{N} k_{ij} \exp\bigl( ( j - i ) I\xi\bigr) \Biggr\vert \Biggr\vert \geq1. \end{aligned}$$

That is, either

$$\begin{aligned} 1- \Biggl\vert r \sum_{j =0}^{N} k_{ij} \exp\bigl( ( j - i ) I\xi\bigr) \Biggr\vert \geq1, \end{aligned}$$

which is impossible to hold, or

$$\begin{aligned} 1- \Biggl\vert r \sum_{j =0}^{N} k_{ij} \exp\bigl( ( j - i ) I\xi\bigr) \Biggr\vert \leq-1, \end{aligned}$$

which is equivalent to \(\vert \sum_{j =0}^{N} k_{ij} \exp ( ( j - i ) I\xi) \vert \geq\frac{2}{r}\).

Thus, the symbol \(\rho ( \xi )\) of the numerical scheme satisfies the von Neumann condition if

$$\begin{aligned} \Biggl\vert \sum_{j =0}^{N} k_{ij} \exp\bigl( ( j - i ) I\xi\bigr) \Biggr\vert \geq \frac{2}{r},\quad \forall i =0,1,2,\dots, N, \end{aligned}$$

which means that the numerical scheme is conditionally stable. This completes the proof of the theorem. □

### Theorem 6

*Numerical Scheme *(37) *is consistent with second*-*order accuracy in space and first*-*order one in time*.

### Proof

Insert \(\psi( x, t )\) into expression (37) to get

$$\begin{aligned} &\frac{\psi ( x_{i}, t_{n +1} ) - \psi( x_{i}, t_{n} )}{\tau}\\ &\quad=\epsilon h^{\alpha-1} \Biggl[ \sum _{j =0}^{i} v_{j}^{\alpha } \psi( x_{i - j}, t_{n +1} ) - \sum_{j =0}^{i +1} v_{j}^{\alpha} \psi( x_{i - j +1}, t_{n +1} ) \Biggr] \\ &\qquad{}+\frac{\rho}{4 h^{2}} \bigl[ \psi( x_{i -2}, t_{n +1} ) -2 \psi( x_{i -2}, t_{n +1} ) + \psi( x_{i +2}, t_{n +1} ) \bigr], \\ &T_{i}^{n} = \frac{\psi ( x_{i}, t_{n +1} ) - \psi( x_{i}, t_{n} )}{\tau} - \epsilon h^{\alpha-1} \Biggl[ \sum_{j =0}^{i} v_{j}^{\alpha} \psi( x_{i - j}, t_{n +1} ) - \sum_{j =0}^{i +1} v_{j}^{\alpha} \psi( x_{i - j +1}, t_{n +1} ) \Biggr] \\ &\phantom{T_{i}^{n} =}{}- \frac{\rho}{4 h^{2}} \bigl[ \psi( x_{i -2}, t_{n +1} ) -2 \psi( x_{i -2}, t_{n +1} ) + \psi( x_{i +2}, t_{n +1} ) \bigr]. \end{aligned}$$

By the control volume averages, combined with a temporal backward difference for the time derivative,

$$\begin{aligned} \frac{d \overline{\psi}_{i} ( t_{n +1} )}{dt} = \frac{\psi ( x_{i}, t_{n +1} ) - \psi ( x_{i}, t_{n} )}{\tau} +\mathcal{O} \bigl( \tau+ h^{2} \bigr), \end{aligned}$$

we have

$$\begin{aligned} T_{i}^{n} = {}& \frac{d \overline{\psi}_{i} ( t_{n +1} )}{dt} -\mathcal{O} \bigl( \tau+ h^{2} \bigr) - \epsilon h^{\alpha-1} \Biggl[\sum_{j =0}^{i} v_{j}^{\alpha} \psi_{i - j}^{n +1} - \sum_{j =0}^{i +1} v_{j}^{\alpha} \psi_{i - j +1}^{n +1} \Biggr] \\ &{}- \frac{v}{4 h^{2}} \bigl[ \psi_{i -2}^{n +1} -2 \psi_{i}^{n +1} + \psi_{i +2}^{n +1} \bigr]. \end{aligned}$$

Substitute (25), (27), and (30) above, it turns to \(T_{i}^{n} =\mathcal{O} ( \tau+ h^{2} ) \). Hence, \(T_{i}^{n} \rightarrow0\) as \(\tau \rightarrow0\) and \(h \rightarrow0\). So, the given numerical scheme is consistent with second-order accuracy in space and first-order in time. This completes the proof of the theorem. □

### Corollary 3

*Numerical Scheme *(37) *is consistent and stable*.

By the fundamental theorem of numerical methods for differential equations, the given numerical scheme is convergent.