# An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fractional Diffusion and Seepage Flow in Homogeneous Media

**Step 1.**Solve the problem in the x-direction (for each fixed pair $({y}_{j},{z}_{k}))$, in order to obtain the intermediate value of the solution, say ${p}_{i,j,k}^{n+1/3}$, from the first equation in (17).

**Step 2.**Solve the problem in the y-direction (for each fixed pair $({x}_{i},{z}_{k}))$, in order to obtain the intermediate value ${p}_{i,j,k}^{n+2/3}$, from the second equation in (17), while using the results that were obtained at Step 1.

**Step 3.**Solve the problem in the z-direction (for each fixed pair $({x}_{i},{y}_{j}))$, from the third equation in (17), using the results of Step 2.

## 3. Theoretical Analysis of the 3D fADI Algorithm

**Theorem**

**1.**

**Proof.**

## 4. The Numerical Implementation and Some Examples

#### 4.1. Some Theoretical Preliminary Considerations

**Step 1.**Compute first the array $\mathrm{\Psi}={({\psi}_{1},{\psi}_{2},{\psi}_{3},{\psi}_{4})}^{T}$, solving the system

**Step 2.**Compute the solution ${p}^{n}$ to a compact difference scheme [28,39,40,41] (see step 3 below), with the four time step sizes $\tau $, $\frac{3}{4}\tau $, $\frac{\tau}{2}$, and $\frac{\tau}{4}$ [40]. This kind of method is usually adopted in order to treat steady convection-diffusion numerical problems on uniform grids [39], rather than time-dependent problems.

**Step 3.**Evaluate the extrapolated solution, ${q}^{n}\left(\tau \right)$, by

#### 4.2. Numerical Examples

**Example 1.**Consider the three dimensional fPDE

**Example 2.**Let us choose, in Equation (30), ($\alpha ,\beta ,\gamma $) = ($1.4,1.5,1.6$), and the (linear) source term $f\left(p\right)=\frac{p}{2}$, instead of the linear source $f(x,y,z,t)$, independent of p, as defined in (31) above. This is a fractional linear reaction–diffusion problem. A forcing term like this often occurs, for instance, in modeling dissolution and precipitation phenomena in porous media [45,46].

**Example 3.**Now consider the linear fractional diffusion equation with an impulsive source,

**Example 4.**We consider the Fisher equation, which is the semilinear (reaction–diffusion) equation [48]

## 5. Conclusions and Future Directions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Classical solution (obtained by a fine grid numerical ADI method with $\tau =h/16$ and $h=1/64$), and (

**b**) exact (analytical) fractional diffusion solution with $(\alpha ,\beta ,\gamma )=(1.4,1.5,1.6)$, for $z=1$ and $T=2$. The same forcing function was used in both cases.

**Figure 2.**Absolute numerical error ${\epsilon}_{N}$ (in log scale) between the exact and the numerical solution of the fractional partial differential equation (fPDE) of Example 1, at $T=2$, with $(\alpha ,\beta ,\gamma )=(1.4,1.5,1.6)$.

**Figure 3.**${L}^{\infty}$ and ${L}^{2}$ discrepancy, again denoted by ${\epsilon}_{N}$ (in log scale) between the numerical solution of the classical problem and that of the fractional problem with $(\alpha ,\beta ,\gamma )$ = $(1.4,1.5,1.6)$, at $T=2$.

**Figure 4.**Numerical absolute errors $\parallel {q}_{fADI}^{n}-{p}^{n}\parallel $ for three values of the time and the space step-sizes, $\tau =h$(solid-red), $\tau =h/2$ (dashed-green), $\tau =h/4$(dashed and dotted-blue), for $\alpha =\beta =\gamma =1.8$, $h=1/N$, on a log-scale.

**Figure 5.**Discrepancy $\parallel {q}_{fADI}^{n}-{q}_{ADI}^{n}\parallel $ for three values of time and space step-sizes, $\tau =h$ (solid-red), $\tau =h/2$ (dashed-green), and $\tau =h/4$ (dashed and dotted-blue), for $\alpha =\beta =\gamma =1.8$, $h=1/N$, on a log-scale.

**Figure 6.**Domain for the variable reaction term coefficient $k(x,y,z)$ divided into the regions ${\mathrm{\Omega}}_{1}^{\prime}$(dotted blue), ${\mathrm{\Omega}}_{3}^{\prime}$ (dotted green), and ${\mathrm{\Omega}}_{2}^{\prime}$(dotted magenta).

**Figure 7.**Solution to Equation (35), at time $T=90$, with the initial condition (36) and parameters $\alpha =\beta =\gamma =2$, $C=0.15,D=0.4,r=0.2$, on the cross section in the plane $(x,y)$, this plane is divided into two regions, the pressure propagates slowly from a region (left) to the other (right) through a silt barrier that links them [48].

**Table 1.**${L}^{\infty}$ and ${L}^{2}$ norm errors (in log scale) and convergence rates for Example 1, when the balanced scheme is used, at time $T=1$, for several values of $(\alpha ,\beta ,\gamma )$, N.

$(\mathit{\alpha},\mathit{\beta},\mathit{\gamma})$ | N | $\parallel {\mathit{q}}^{\mathit{n}}-{\mathit{p}}^{\mathit{n}}{\parallel}_{\mathit{\infty}}$ | ${\mathit{r}}_{\mathit{space}}$ | $\parallel {\mathit{q}}^{\mathit{n}}-{\mathit{p}}^{\mathit{n}}{\parallel}_{2}$ | ${\mathit{r}}_{\mathit{space}}$ | ${\mathit{r}}_{\mathit{time}}$ |
---|---|---|---|---|---|---|

$(1.2,1.2,1.2)$ | 8 | $1.5023\times {10}^{-2}$ | $--$ | $7.5264\times {10}^{-2}$ | $--$ | $--$ |

16 | $2.8925\times {10}^{-3}$ | $2.002$ | $1.6581\times {10}^{-2}$ | $2.015$ | $3.009$ | |

32 | $6.8952\times {10}^{-4}$ | $2.000$ | $5.3265\times {10}^{-3}$ | $2.007$ | $3.006$ | |

64 | $2.3561\times {10}^{-4}$ | $1.996$ | $2.3654\times {10}^{-4}$ | $1.999$ | $3.001$ | |

$(1.4,1.5,1.6)$ | 8 | $2.0253\times {10}^{-2}$ | $--$ | $3.2564\times {10}^{-2}$ | $--$ | $--$ |

16 | $2.9541\times {10}^{-3}$ | 2.001 | $3.1254\times {10}^{-2}$ | $2.011$ | $3.010$ | |

32 | $7.1254\times {10}^{-4}$ | $2.000$ | $7.9856\times {10}^{-3}$ | $2.007$ | $3.005$ | |

64 | $2.5648\times {10}^{-4}$ | $1.988$ | $2.8930\times {10}^{-2}$ | $1.998$ | $2.998$ | |

$(1.9,1.9,1.9)$ | 8 | $2.324\times {10}^{-2}$ | $--$ | $3.5852\times {10}^{-2}$ | $--$ | $--$ |

16 | $3.5984\times {10}^{-3}$ | $2.005$ | $3.9852\times {10}^{-2}$ | $2.009$ | $3.002$ | |

32 | $7.5214\times {10}^{-4}$ | 2.002 | $7.7815\times {10}^{-3}$ | $2.008$ | $3.001$ | |

64 | $2.8594\times {10}^{-4}$ | 1.992 | $3.2852\times {10}^{-4}$ | $1.989$ | $3.000$ | |

$(2.0,2.0,2.0)$ | 8 | $5.2154\times {10}^{-3}$ | $--$ | $6.7952\times {10}^{-3}$ | $--$ | $--$ |

16 | $6.7854\times {10}^{-4}$ | $2.001$ | $7.1248\times {10}^{-4}$ | $2.001$ | $3.000$ | |

32 | $5.7453\times {10}^{-5}$ | $1.999$ | $7.9173\times {10}^{-5}$ | $1.998$ | $3.000$ | |

64 | $6.7852\times {10}^{-5}$ | $2.000$ | $7.7945\times {10}^{-5}$ | $2.000$ | $2.999$ |

**Table 2.**${L}^{\infty}$ and ${L}^{2}$ norm errors, and convergence rates for Example 1, when the balanced scheme is used, at time $T=2$, for several values of N.

$(\mathit{\alpha},\mathit{\beta},\mathit{\gamma})$ | N | $\parallel {\mathit{q}}^{\mathit{n}}-{\mathit{p}}^{\mathit{n}}{\parallel}_{\mathit{\infty}}$ | ${\mathit{r}}_{\mathit{space}}$ | $\parallel {\mathit{q}}^{\mathit{n}}-{\mathit{p}}^{\mathit{n}}{\parallel}_{2}$ | ${\mathit{r}}_{\mathit{space}}$ | ${\mathit{r}}_{\mathit{time}}$ |
---|---|---|---|---|---|---|

$(1.4,1.5,1.6)$ | 8 | $8.2654\times {10}^{-3}$ | $--$ | $9.7452\times {10}^{-3}$ | $--$ | $--$ |

16 | $2.6584\times {10}^{-4}$ | $2.001$ | $2.7852\times {10}^{-3}$ | $2.001$ | $3.002$ | |

32 | $7.0145\times {10}^{-5}$ | $2.000$ | $5.3255\times {10}^{-4}$ | $1.999$ | $3.000$ | |

64 | $1.2420\times {10}^{-5}$ | $1.999$ | $4.7852\times {10}^{-5}$ | $2.000$ | $2.999$ |

**Table 3.**${L}^{\infty}$ and ${L}^{2}$ norm discrepancy for Example 1, when the balanced scheme is used, at time $T=1$, for several values of N.

$(\mathit{\alpha},\mathit{\beta},\mathit{\gamma})$ | N | $\parallel {\mathit{q}}_{\mathit{fADI}}^{\mathit{n}}-{\mathit{q}}_{\mathit{ADI}}^{\mathit{n}}{\parallel}_{\mathit{\infty}}$ | $\parallel {\mathit{q}}_{\mathit{fADI}}^{\mathit{n}}-{\mathit{q}}_{\mathit{ADI}}^{\mathit{n}}{\parallel}_{2}$ |
---|---|---|---|

$(1.2,1.2,1.2)$ | 256 | $6.1758\times {10}^{-7}$ | $3.8458\times {10}^{-6}$ |

512 | $2.2548\times {10}^{-7}$ | $7.8442\times {10}^{-7}$ | |

$(1.4,1.5,1.6)$ | 256 | $2.8287\times {10}^{-7}$ | $1.1205\times {10}^{-6}$ |

512 | $3.7852\times {10}^{-8}$ | $3.7854\times {10}^{-7}$ | |

$(1.9,1.9,1.9)$ | 256 | $1.1582\times {10}^{-7}$ | $5.7852\times {10}^{-7}$ |

512 | $1.7855\times {10}^{-8}$ | $5.0023\times {10}^{-7}$ | |

$(2.0,2.0,2.0)$ | 256 | $8.1158\times {10}^{-8}$ | $8.7852\times {10}^{-7}$ |

512 | $1.7852\times {10}^{-8}$ | $6.7852\times {10}^{-8}$ |

**Table 4.**${L}^{\infty}$ and ${L}^{2}$ norm discrepancy for Example 1, when the balanced scheme is used, at time $T=2$, for several values of N.

$(\mathit{\alpha},\mathit{\beta},\mathit{\gamma})$ | N | $\parallel {\mathit{q}}_{\mathit{fADI}}^{\mathit{n}}-{\mathit{q}}_{\mathit{ADI}}^{\mathit{n}}{\parallel}_{\mathit{\infty}}$ | $\parallel {\mathit{q}}_{\mathit{fADI}}^{\mathit{n}}-{\mathit{q}}_{\mathit{ADI}}^{\mathit{n}}{\parallel}_{2}$ |
---|---|---|---|

$(1.4,1.5,1.6)$ | 256 | $5.7854\times {10}^{-7}$ | $1.4002\times {10}^{-8}$ |

512 | $2.7584\times {10}^{-8}$ | $7.7852\times {10}^{-8}$ |

**Table 5.**CPU times in clocks ticks units $\left(CT\right)$ for the unbalanced and the balanced versions of the fADI scheme, $C{T}_{bal}$, and $C{T}_{unbal}$, respectively, required attaining an error of order ${10}^{-5}$, for $h=1/100$ and several values of the fractional orders, $\alpha ,\beta ,\gamma $, and the advection coefficients, ${d}_{x},{d}_{y},{d}_{z}$.

$(\mathit{\alpha},\mathit{\beta},\mathit{\gamma})$ | $({\mathit{d}}_{\mathit{x}},{\mathit{d}}_{\mathit{y}},{\mathit{d}}_{\mathit{z}})$ | ${\mathit{CT}}_{\mathit{unbal}}$ | ${\mathit{CT}}_{\mathit{bal}}$ |
---|---|---|---|

$(1.8,1.5,1.2)$ | $(10,5,1)$ | $2.256\times {10}^{-3}$ | $8.236\times {10}^{-2}$ |

$(1.8,1.5,1.3)$ | $(8,4,1)$ | $4.365\times {10}^{-3}$ | $4.255\times {10}^{-2}$ |

$(1.8,1.6,1.4)$ | $(7,3,1)$ | $3.256\times {10}^{-2}$ | $5.778\times {10}^{-2}$ |

$(1.8,1.6,1.5)$ | $(5,2,1)$ | $7.289\times {10}^{-2}$ | $2.389\times {10}^{-3}$ |

$(1.8,1.7,1.6)$ | $(3,1,1)$ | $6.258\times {10}^{-1}$ | $1.756\times {10}^{-3}$ |

$(1.8,1.7,1.7)$ | $(2,1,1)$ | $8.236\times {10}^{-1}$ | $1.586\times {10}^{-3}$ |

$(1.8,1.8,1.8)$ | $(1,1,1)$ | $9.266\times {10}^{-1}$ | $1.256\times {10}^{-3}$ |

**Table 6.**CPU times in clocks ticks units $\left(CT\right)$ for the unbalanced and the balanced versions of the fractional Alternating Direction Implicit (fADI) scheme, $C{T}_{bal}$, and $C{T}_{unbal}$, respectively, required attaing an error of order ${10}^{-5}$, for $N=100$ and several values $h=\frac{1}{\gamma N}$, and of the fractional orders, $\alpha ,\beta ,\gamma $.

$(\mathit{\alpha},\mathit{\beta},\mathit{\gamma})$ | h | ${\mathit{CT}}_{\mathit{unbal}}$ | ${\mathit{CT}}_{\mathit{bal}}$ |
---|---|---|---|

$(1.8,1.5,1.2)$ | $0.08$ | $5.256\times {10}^{-3}$ | $7.256\times {10}^{-2}$ |

$(1.8,1.5,1.3)$ | $0.08$ | $6.001\times {10}^{-2}$ | $3.856\times {10}^{-2}$ |

$(1.8,1.6,1.4)$ | $0.07$ | $7.898\times {10}^{-2}$ | $4.258\times {10}^{-2}$ |

$(1.8,1.6,1.5)$ | $0.07$ | $8.856\times {10}^{-2}$ | $1.759\times {10}^{-2}$ |

$(1.8,1.7,1.6)$ | $0.07$ | $4.125\times {10}^{-1}$ | $1.896\times {10}^{-2}$ |

$(1.8,1.7,1.7)$ | $0.06$ | $7.156\times {10}^{-1}$ | $1.325\times {10}^{-3}$ |

$(1.8,1.8,1.8)$ | $0.05$ | $8.780\times {10}^{-1}$ | $1.126\times {10}^{-3}$ |

**Table 7.**CPU times in clocks ticks units $\left(CT\right)$ for the unbalanced and the balanced versions of the fADI scheme, $C{T}_{bal}$, and $C{T}_{unbal}$, respectively, required attaining an error of order ${10}^{-5}$, for $h=1/100$ and several values of the fractional orders, $\alpha ,\beta ,\gamma $, and the advection coefficients ${d}_{x},{d}_{y},{d}_{z}$.

$(\mathit{\alpha},\mathit{\beta},\mathit{\gamma})$ | $({\mathit{d}}_{\mathit{x}},{\mathit{d}}_{\mathit{y}},{\mathit{d}}_{\mathit{z}})$ | ${\mathit{CT}}_{\mathit{unbal}}$ | ${\mathit{CT}}_{\mathit{bal}}$ |
---|---|---|---|

$(1.5,1.8,1.2)$ | $(5,10,1)$ | $7.152\times {10}^{-2}$ | $8.256\times {10}^{-2}$ |

$(1.5,1.8,1.3)$ | $(4,8,1)$ | $9.856\times {10}^{-2}$ | $7.026\times {10}^{-2}$ |

$(1.6,1.8,1.4)$ | $(3,7,1)$ | $6.856\times {10}^{-1}$ | $5.785\times {10}^{-2}$ |

$(1.6,1.8,1.5)$ | $(2,5,1)$ | $7.019\times {10}^{-1}$ | $3.256\times {10}^{-3}$ |

$(1.7,1.8,1.6)$ | $(1,3,1)$ | $8.786\times {10}^{-1}$ | $1.756\times {10}^{-3}$ |

$(1.7,1.8,1.7)$ | $(1,2,1)$ | $9.325\times {10}^{-1}$ | $1.586\times {10}^{-3}$ |

$(1.8,1.8,1.8)$ | $(1,1,1)$ | $1.256$ | $1.025\times {10}^{-3}$ |

**Table 8.**CPU times in clocks ticks units $\left(CT\right)$ for the unbalanced and the balanced versions of the fADI scheme, $C{T}_{bal}$, and $C{T}_{unbal}$, respectively, required attaining an error of order ${10}^{-5}$, for $N=100$ and several values $h=\frac{1}{\gamma N}$, and of the fractional orders, $\alpha ,\beta ,\gamma $.

$(\mathit{\alpha},\mathit{\beta},\mathit{\gamma})$ | h | ${\mathit{CT}}_{\mathit{unbal}}$ | ${\mathit{CT}}_{\mathit{bal}}$ |
---|---|---|---|

$(1.5,1.8,1.2)$ | $0.08$ | $7.266\times {10}^{-2}$ | $5.125\times {10}^{-2}$ |

$(1.5,1.8,1.3)$ | $0.08$ | $9.014\times {10}^{-2}$ | $4.856\times {10}^{-2}$ |

$(1.6,1.8,1.4)$ | $0.07$ | $5.888\times {10}^{-1}$ | $2.558\times {10}^{-2}$ |

$(1.6,1.8,1.5)$ | $0.07$ | $7.896\times {10}^{-1}$ | $1.325\times {10}^{-3}$ |

$(1.7,1.8,1.6)$ | $0.07$ | $3.965\times {10}^{-1}$ | $1.756\times {10}^{-3}$ |

$(1.7,1.8,1.7)$ | $0.06$ | $7.156$ | $1.256\times {10}^{-3}$ |

$(1.8,1.8,1.8)$ | $0.05$ | $8.780$ | $9.254\times {10}^{-4}$ |

**Table 9.**CPU times in clocks ticks units $\left(CT\right)$ for the unbalanced and the balanced versions of the fADI scheme, $C{T}_{bal}$, and $C{T}_{unbal}$, respectively, required attaining an error of order ${10}^{-5}$, for $h=1/100$, ${d}_{x}={d}_{y}={d}_{z}=0$, and several values of the fractional orders, $\alpha ,\beta ,\gamma $.

$(\mathit{\alpha},\mathit{\beta},\mathit{\gamma})$ | ${\mathit{CT}}_{\mathit{unbal}}$ | ${\mathit{CT}}_{\mathit{bal}}$ |
---|---|---|

$(1.5,1.5,1.5)$ | $8.458$ | $7.758\times {10}^{-4}$ |

$(1.3,1.8,1.3)$ | $2.786\times {10}^{-1}$ | $7.256\times {10}^{-2}$ |

$(1.2,1.5,1.2)$ | $6.289\times {10}^{-1}$ | $2.389\times {10}^{-3}$ |

$(1.5,1.7,1.5)$ | $6.258\times {10}^{-1}$ | $1.756\times {10}^{-3}$ |

$(1.7,1.8,1.7)$ | $3.276\times {10}^{-1}$ | $5.785\times {10}^{-3}$ |

$(1.8,1.8,1.8)$ | $1.256\times $ | $8.785\times {10}^{-4}$ |

**Table 10.**${L}^{2}$ norm discrepancies, and convergence rates for Example 2, when the balanced scheme is used, at time $T=2$, source term $f\left(p\right)=\frac{p}{2}$, $(\alpha ,\beta ,\gamma )$ = $(1.1,1.7,1)$, $h=1/N$, and several values of N.

N | $\mathit{\tau}$ | $\parallel {\mathit{q}}_{\mathit{fADI}}^{\mathit{n}}-{\mathit{q}}_{\mathit{ADI}}^{\mathit{n}}{\parallel}_{2}$ | ${\mathit{Rate}}_{\mathit{space}}$ | ${\mathit{Rate}}_{\mathit{time}}$ |
---|---|---|---|---|

16 | $h/16$ | $8.5825\times {10}^{-4}$ | $--$ | $--$ |

32 | $h/16$ | $3.8501\times {10}^{-4}$ | $2.000$ | $3.001$ |

64 | $h/16$ | $1.7854\times {10}^{-5}$ | $2.000$ | $3.000$ |

**Table 11.**${L}^{2}$ norm errors, discrepancies and convergence rates for Example 3, when the balanced scheme is used, with $\alpha =\beta =\gamma =1.8$, at time $T=1$, for $h=1/N$ and several values of N and $\tau $. The two numbers in the rates columns refer to the first and second algorithms, respectively.

N | $\mathit{\tau}$ | $\parallel {\mathit{q}}_{\mathit{fADI}}^{\mathit{n}}-{\mathit{p}}^{\mathit{n}}{\parallel}_{2}$ | $\parallel {\mathit{q}}_{\mathit{fADI}}^{\mathit{n}}-{\mathit{q}}_{\mathbf{ADI}}^{\mathit{n}}{\parallel}_{2}$ | Space Conv. Rates | Time Conv. Rates |
---|---|---|---|---|---|

16 | h | $2.0122\times {10}^{-1}$ | $2.5684$ | – | – |

32 | h | $3.8524\times {10}^{-2}$ | $1.8524\times {10}^{-1}$ | – | – |

64 | h | $2.1852\times {10}^{-3}$ | $3.2658\times {10}^{-2}$ | – | – |

16 | $h/2$ | $5.7616\times {10}^{-1}$ | $1.0226$ | 1.96–1.82 | 2.98–2.92 |

32 | $h/2$ | $3.1658\times {10}^{-2}$ | $2.5802\times {10}^{-1}$ | 1.99–1.93 | 2.99–2.95 |

64 | $h/2$ | $3.5800\times {10}^{-3}$ | $6.3552\times {10}^{-2}$ | 1.98–1.89 | 2.99–2.96 |

16 | $h/4$ | $1.7525\times {10}^{-2}$ | $3.2753$ | 2.00–1.96 | 3.00–2.98 |

32 | $h/4$ | $2.1358\times {10}^{-3}$ | $2.1258\times {10}^{-1}$ | 2.00–1.98 | 3.00–2.99 |

64 | $h/4$ | $3.3654\times {10}^{-4}$ | $9.4528\times {10}^{-2}$ | 2.00–2.00 | 3.00–3.01 |

**Table 12.**Computational times (in seconds) for the unbalanced and balanced version of the fADI scheme, respectively, ${T}_{unbal}$ and ${T}_{bal}$, required to attain an accuracy of order ${10}^{-2}$, with $(\alpha ,\beta ,\gamma )=(1.8,1.6,1.4)$, at time $T=1$, for $h=1/N$, and several values of N and $\tau $.

N | $\mathit{\tau}$ | ${\mathit{T}}_{\mathit{unbal}}$ | ${\mathit{T}}_{\mathit{bal}}$ |
---|---|---|---|

16 | h | $55.8$ | $33.9$ |

32 | h | $111.1$ | $67.8$ |

64 | h | $223.4$ | $135.7$ |

16 | $h/2$ | $117.2$ | $71.2$ |

32 | $h/2$ | $233.3$ | $142.4$ |

64 | $h/2$ | $472.3$ | $285.0$ |

16 | $h/4$ | $234.4$ | $142.4$ |

32 | $h/4$ | $466.7$ | $284.8$ |

64 | $h/4$ | $938.3$ | $570.0$ |

**Table 13.**CPU times in clocks ticks units $\left(CT\right)$, and memory ($RAM$ in $MB$) for the unbalanced and the balanced versions of the fADI scheme, $C{T}_{bal},RA{M}_{bal}$ and $C{T}_{unbal},RA{M}_{unbal}$, respectively, with $(\alpha ,\beta ,\gamma )=(1.8,1.6,1.4)$, at time $T=1$, for $h=1/N$, and several values of N and $\tau $.

N | $\mathit{\tau}$ | ${\mathit{CT}}_{\mathit{bal}}$ | ${\mathit{RAM}}_{\mathit{bal}}$ | ${\mathit{CT}}_{\mathit{unbal}}$ | ${\mathit{RAM}}_{\mathit{unbal}}$ |
---|---|---|---|---|---|

16 | h | $1.944\times {10}^{-3}$ | $0.15$ | $2.778\times {10}^{-3}$ | $0.23$ |

32 | h | $4.416\times {10}^{-3}$ | $0.23$ | $5.278\times {10}^{-3}$ | $0.34$ |

64 | h | $8.889\times {10}^{-2}$ | $0.81$ | $1.027\times {10}^{-2}$ | $0.92$ |

16 | $h/2$ | $3.050\times {10}^{-2}$ | $0.24$ | $3.611\times {10}^{-3}$ | $0.33$ |

32 | $h/2$ | $5.555\times {10}^{-3}$ | $0.42$ | $6.944\times {10}^{-3}$ | $0.51$ |

64 | $h/2$ | $1.028\times {10}^{-2}$ | $1.23$ | $1.361\times {10}^{-2}$ | $1.38$ |

16 | $h/4$ | $5.278\times {10}^{-3}$ | $0.59$ | $6.389\times {10}^{-3}$ | $0.67$ |

32 | $h/4$ | $1.139\times {10}^{-2}$ | $0.83$ | $1.310\times {10}^{-3}$ | $0.95$ |

64 | $h/4$ | $2.778\times {10}^{-2}$ | $1.70$ | $2.667\times {10}^{-2}$ | $1.92$ |

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**MDPI and ACS Style**

Concezzi, M.; Spigler, R.
An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations. *Fractal Fract.* **2020**, *4*, 57.
https://doi.org/10.3390/fractalfract4040057

**AMA Style**

Concezzi M, Spigler R.
An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations. *Fractal and Fractional*. 2020; 4(4):57.
https://doi.org/10.3390/fractalfract4040057

**Chicago/Turabian Style**

Concezzi, Moreno, and Renato Spigler.
2020. "An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations" *Fractal and Fractional* 4, no. 4: 57.
https://doi.org/10.3390/fractalfract4040057