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In this research article, two finite difference implicit numerical schemes are described to approximate the numerical solution of the two - dimension modified reaction diffusion Fisher ’ s system which exists in coupled form. Finite difference implicit schemes show unconditionally stable and second - order accurate nature of computational algorithm also the validation and comparison of analytical solution, are done through the examples having known analytical solution. It is found that the numerical schemes are in excellent agreement with the analytical solution. We found, second - implicit scheme is much faster than the first with good rate of convergence also we used NVIDA devices to accelerate the computations and efficiency of the algorithm. Numerical results show our proposed schemes with use of HPC (High performance computing) are very efficient and reliable.

Reaction diffusion (RD) equations rise up naturally in systems consisting of many interacting factors, such as chemical reactions and are widely used to identify pattern formation phenomena in diverseness of biological and physical systems [

∂ t u = β ( ∇ 2 u ) + R ( u ) , (1)

where u = u ( x , y , t ) is a vector of concentration variables, R(u) describes a local reaction kinetics and the Laplace operator ∇ 2 acts on the vector u component wise, also b denotes a diagonal diffusion coefficient matrix [

∂ t u ( x , y , t ) = β ( u x x + u y y ) + R ( u ) (2)

with a nonlinear source term R ( u ) = u − u 2 [

The reaction diffusion Equation (2) represents a model equation for the evolution of a neutron population in a nuclear reactor and also arises in chemical engineering applications, such equation allows for the effects of linear diffusion by means of u x x + u y y and nonlinear local multiplication or reaction through R ( u ) [

Researchers have studied these model problems such as the stability of symmetric traveling waves in the Cauchy problem for a more general case than Equation (2); also some researchers explained perturbation method and found an approximate solution by expanding the solution in terms of a power series and in terms of some small parameters [

In this paper, we suggest a reaction diffusion system, which agree to several physical phenomena, the most common is the change in space and time of the concentration of one or more chemical substances. Local chemical reactions in which the metamorphosed into each other, and diffusion which causes the substances to spread out over a surface in space. Reaction diffusion systems are naturally applied in chemistry. However, the system can also describe dynamical processes of non-chemical nature. Mathematically, reaction diffusion systems take the form of semilinear parabolic partial differential equations. The general form of our proposed reaction diffusion system in two dimension, is

u t = β ( u x x + u y y ) + u 2 v − α u , ( x , y ) ∈ ( − ∞ , ∞ ) , t ≥ 0 (3)

v t = β ( v x x + v y y ) − v 2 u , ( x , y ) ∈ ( − ∞ , ∞ ) , t ≥ 0 (4)

where b is diffusion coefficient and a is a reactive factor, and u ( x , t ) is concentration and v ( x , t ) is the velocity of the chemical reaction. The aim of this work is to look into the viability of finite difference schemes for the numerical solution of two-dimension coupled reaction diffusion system. The proposed finite difference schemes show good agreement with analytical solution along efficiency in time. Comparison of two finite difference (FD) schemes is also mentioned with CPU efficiency.

The outlook of the paper is in Section 2 analytical solution, Section 3 smoothness and uniqueness, Section 4 numerical methods, Section 5 numerical results and Section 6 discussion.

To derive the analytical solution of the given system in (3), (4), we assume the solution of the two dimension coupled reaction diffusion system, in the following form

u ( x , y , t ) = e α t 2 − x − y (5)

v ( x , y , t ) = e − α t 2 + x + y (6)

In order to guarantee the smoothness and uniqueness of a positive solution and to obtain upper and lower bounds of the solution, it is necessary to impose some general assumptions on the various physical parameters and the reaction function [

u t − ∇ ⋅ ( β ∇ u ) = − u m f ( u , v ) v t − ∇ ⋅ ( β ∇ v ) = u m g ( u , v ) } where m ≥ 1 , f ( u , v ) = u v − α and g ( u , v ) = v 2 B 1 [ u ] = h 1 ( x , y ) B 2 [ v ] = h 2 ( x , y ) } Boundary Conditions , t > 0 , ( x , y ) ∈ Ω u ( 0 , x , y ) = u 0 ( x , y ) v ( 0 , x , y ) = v 0 ( x , y ) } Initial Conditions , ( x , y ) ∈ Ω } (7)

motivated by the nonlinear reaction functions given by Equation (7), we make the following basic assumption on functions f and g [

∂ f / ∂ v exists and is bounded subsets of domain W and there exists a function with c o ( x , y ) ≥ 0 , such that 0 ≤ f ( u , v 1 ) ≤ f ( u , v 2 ) ≤ c o ( x , y ) for 0 ≤ v 1 ≤ v 2 ≤ ∞ [

F 1 ( x , y , u , v ) = − u m f ( u , v ) F 2 ( x , y , u , v ) = + u m g ( u , v ) , } (8)

where above Equation (8) represents F 1 , F 2 are quasi monotone non-increasing and quasi monotone non-decreasing functions in W respectively [

A smooth pair of two vector functions ( u ⌣ , v ⌣ ) , ( u ¯ , v ¯ ) defined in R + × Ω are called upper and lower solutions respectively, if they satisfy the following inequalities

u ⌣ t − ∇ ⋅ ( β ∇ u ⌣ ) + u ⌣ m f ( x , v ¯ ) ≥ 0 ≥ u ¯ t − ∇ ⋅ ( β ∇ u ¯ ) + u ¯ m f ( x , y , v ⌣ ) v ⌣ t − ∇ ⋅ ( β ∇ v ⌣ ) − u ⌣ m g ( x , v ⌣ ) ≥ 0 ≥ v ¯ t − ∇ ⋅ ( β ∇ v ¯ ) − u ⌣ m g ( x , y , v ¯ ) } where m ≥ 1 , f ( x , y , v ) = u v ⌣ − α and g ( x , y , v ) = v ¯ 2 B 1 [ u ⌣ ] ≥ h 1 ( x , y ) ≥ B 1 [ u ¯ ] B 2 [ v ⌣ ] ≥ h 2 ( x , y ) ≥ B 2 [ u ¯ ] } Boundary Conditions , t > 0 , ( x , y ) ∈ Ω u ⌣ ( 0 , x , y ) ≥ u 0 ( x , y ) ≥ u ¯ ( 0 , x , y ) v ⌣ ( 0 , x , y ) ≥ v 0 ( x , y ) ≥ v ¯ ( 0 , x , y ) } Initial Conditions , ( x , y ) ∈ Ω } (9)

In the above definitions the smoothness of ( u ⌣ , v ⌣ ) , ( u ¯ , v ¯ ) is in the sense that these functions are continuously differentiable to the order appeared in Equations (7) and (8) respectively [

Let f and g satisfy above hypothesis (H). If there exist upper and lower solutions ( u ⌣ , v ⌣ ) , ( u ¯ , v ¯ ) of system 7, such that u ⌣ ≤ u ¯ and v ⌣ ≤ v ¯ in R + × Ω , then the sequence { u ˙ k , v ˙ k } , { u _ k , v _ k } converges monotonically from above and below, respectively, to a unique solution (u, v) of system (7) [

u ⌣ ( t , x , y ) ≤ u ( t , x , y ) ≤ u ¯ ( t , x , y ) v ⌣ ( t , x , y ) ≤ v ( t , x , y ) ≤ v ¯ ( t , x , y ) } t > 0 , ( x , y ) ∈ Ω (10)

The usefulness of the above theorem is that through suitable construction of upper and lower solutions, not only can the existence problem be ensured, but the stability and the asymptotic behavior of the time dependent solution can also be established from the behavior of the upper and lower solutions [

We consider the numerical solution of the nonlinear system in (5), (6) and (7) in a finite domain Ω = { ( x , y ) | a < x < b , c < y < d } , where the first step is to choose integers n and m to define step sizes h = ( b − a ) / n and k = ( d − c ) / m [

The Crank Nicolson scheme for the system in (3) and (4) can be displayed as follows:

u ^ = u l , m n + 1 + u l , m n 2 v ^ = v l , m n + 1 + v l , m n 2 δ x 2 u ^ = u ^ l + 1 , m − 2 u ^ l , m + u ^ l − 1 , m h 2 δ y 2 u ^ = u ^ l , m + 1 − 2 u ^ l , m + u ^ l , m − 1 h 2 δ x 2 v ^ = v ^ l + 1 , m − 2 v ^ l , m + v ^ l − 1 , m h 2 δ y 2 v ^ = v ^ l , m + 1 − 2 v ^ l , m + v ^ l , m − 1 h 2 u l , m n + 1 − u l , m n − R 1 ( δ x 2 + δ y 2 ) u ^ − k 8 ( v ^ u ^ 2 ) + 0.5 k ( u ^ ) = 0 v l , m n + 1 − v l , m n − R 1 ( δ x 2 + δ y 2 ) v ^ + k 8 ( v ^ 2 u ^ ) = 0 , } (11)

where R 1 = k β h 2 [

scheme with block linear penta diagonal structure [

In search of a time efficient alternate, we analyzed the naive version of the Crank Nicolson scheme for the two dimensional equation, and find out that that scheme is not time efficient such that to get time efficiency, the common name of Alternating Direction Implicit (ADI) method can be used [

( 1 − r 2 δ x 2 ) u ^ l , m = ( 1 + r 2 δ y 2 ) u l , m n + Δ t f l , m n ( 1 − r 2 δ y 2 ) u l , m n + 1 = ( 1 + r 2 δ x 2 ) u ^ l , m + Δ t f ^ l , m } f l , m n = u l , m n ( u l , m n v l , m n − α ) ( 1 − r 2 δ x 2 ) v ^ l , m = ( 1 + r 2 δ y 2 ) v l , m n + Δ t g l , m n ( 1 − r 2 δ y 2 ) v l , m n + 1 = ( 1 + r 2 δ x 2 ) v ^ l , m + Δ t g ^ l , m } g l , m n = − u l , m n ( v l , m n v l , m n ) } (12)

The trick used in constructing of ADI scheme, is to split time step into two sweeps and apply two different stencils in each half time step, therefore to increment time by one time step in grid point , we first compute both of these stencils, such that the resulting linear system is block tridiagonal [

The nonlinear system of Equation (12), can be written in the form:

R ( W ) = 0 , (13)

where R = ( r 1 , r 2 , r 3 , ⋯ , r 2 n ) t , W = ( u 1 n + 1 , v 1 n + 1 , u 2 n + 1 , v 2 n + 1 , ⋯ , u m n + 1 , v m n + 1 ) and r 1 , r 2 , r 3 , ⋯ , r 2 n are the nonlinear equations obtained from the system in Equation (12). The system of Equations in (12) is solved by Newton's iterative method using the following steps:

1) Specify W ( 0 ) as an initial approximation.

2) For k = 0 , 1 , 2 , ⋯ until convergence achieve.

- Solve the linear system A ( W ( k ) ) Δ W ( k ) = − R ( W (k) )

- Specify W ( k + 1 ) = W ( k ) + Δ W ( k ) ,

where A ( W ( k ) ) is ( m × m ) Jacobian matrix, which is computed analytically and Δ W ( k ) is the correction vector [

stopped when ‖ R ( W ( k ) ) ‖ ∞ ≤ T o l with Tol is a very small prescribed value. The

linear system obtained from Newton's iterative method, is solved by Gauss elimination method with partial pivoting also convergence done with iterations along less CPU time [

The accuracy and consistency of the schemes is measured in terms of error norms specially L 2 and L ∞ which are defined as:

L ∞ = ‖ u ecact − u Approximation ‖ ∞ = max 1 ≤ i ≤ m ∑ j = 1 m | u j ecact − u j Approximation | L 2 = ‖ u ecact − u Approximation ‖ 2 = ∑ j = 1 m | u j ecact − u j Approximation | Rate = log ( Error h / Error h / 2 ) log ( h / ( h / 2 ) ) } (14)

Two more interesting error are listed below,

Error relative = ∑ i ∑ j | u i , j ecact − u i , j Approximation | 2 ∑ j | u i , j exact | 2 RMS = ∑ i ∑ j ( u i , j ecact − u i , j Approximation ) 2 M 2 , where M = Total terms } (15)

Numerical computations have been performed using the uniform grid, for the test problem, the approximated and analytical solutions such as u ( x , y , t ) and U ( x , y , t ) have been given in

Grid Size | Error relative | RMS | L 2 | L ∞ |
---|---|---|---|---|

11 ´ 11 | 0.0089 | 0.0152 | 0.1671 | 0.0285 |

21 ´ 21 | 0.0091 | 0.0312 | 0.1869 | 0.0463 |

31 ´ 31 | 0.0091 | 0.0403 | 0.2139 | 0.0512 |

41 ´ 41 | 0.0095 | 0.0679 | 0.3913 | 0.0710 |

51 ´ 51 | 0.0099 | 0.0931 | 0.5524 | 0.0989 |

Grid Size | Rate L 2 | Rate L ∞ | L 2 | L ∞ |
---|---|---|---|---|

11 ´ 11 | 2.9385 | 2.0235 | 0.1671 | 0.0285 |

21 ´ 21 | 2.2367 | 1.9896 | 0.1869 | 0.0463 |

31 ´ 31 | 1.8976 | 1.4781 | 0.2139 | 0.0512 |

41 ´ 41 | 1.2145 | 1.2797 | 0.3913 | 0.0710 |

51 ´ 51 | 1.1135 | 1.0923 | 0.5524 | 0.0989 |

Space Location | u a p p . | U Analytical | Error u − U |
---|---|---|---|

(0.0152, 0.1667) | 0.494155003687347 | 0.494367466355141 | 0.00021246 |

(0.0455, 0.1970) | 0.53835177275621 | 0.535255683530614 | 0.0031 |

(0.0606, 0.2424) | 0.565563567758470 | 0.558073805129842 | 0.0075 |

(0.1970, 0.1970) | 0.626597867998406 | 0.621185376572772 | 0.0054 |

( 0.2879, 0.2879) | 0.749945814748917 | 0.739953358282411 | 0.0100 |

Grid Size | Error relative | RMS | L 2 | L ∞ |
---|---|---|---|---|

11 ´ 11 | 0.0741 | 0.0173 | 0.1899 | 0.0713 |

21 ´ 21 | 0.0341 | 0.0283 | 0.5934 | 0.0428 |

31 ´ 31 | 0.0123 | 0.0362 | 1.1231 | 0.0232 |

41 ´ 41 | 0.0099 | 0.0428 | 1.7538 | 0.0155 |

51 ´ 51 | 0.0090 | 0.0485 | 2.4713 | 0.0136 |

t = time | Error relative | RMS | L 2 | L ∞ |
---|---|---|---|---|

0.01 | 0.00013 | 0.0102 | 1.04713 | 0.00198 |

0.05 | 0.00089 | 0.0209 | 1.09513 | 0.00589 |

0.1 | 0.0011 | 0.0319 | 1.12686 | 0.00989 |

0.5 | 0.0039 | 0.0400 | 1.8535 | 0.0109 |

1 | 0.0090 | 0.0485 | 2.4713 | 0.0136 |

k = time steps | Error relative | RMS | L 2 | L ∞ |
---|---|---|---|---|

0.01 | 0.0484 | 0.2942 | 2.3542 | 0.1236 |

0.001 | 0.0483 | 0.2936 | 2.3393 | 0.1240 |

0.0001 | 0.0461 | 0.2935 | 2.3380 | 0.1241 |

0.00001 | 0.0423 | 0.2935 | 2.3378 | 0.1241 |

Grid Size | Rate L 2 | Rate L ∞ | L 2 | L ∞ |
---|---|---|---|---|

11 ´ 11 | 2.8009 | 2.8201 | 0.1899 | 0.0113 |

21 ´ 21 | 2.8458 | 2.8733 | 0.5934 | 0.0128 |

31 ´ 31 | 2.8619 | 2.8934 | 1.1231 | 0.0132 |

51 ´ 51 | 2.8458 | 2.8733 | 2.4713 | 0.0136 |

h | Error relative | RMS | L 2 | L ∞ |
---|---|---|---|---|

0.04 | 0.0483 | 0.2936 | 7.3393 | 0.1240 |

0.02 | 0.0118 | 0.0406 | 1.0159 | 0.0168 |

0.01 | 0.0107 | 0.0357 | 0.8920 | 0.0148 |

0.005 | 0.0072 | 0.0219 | 0.5484 | 0.0090 |

0.000625 | 0.0002 | 0.0118 | 0.0253 | 0.0011 |

Grid Size | Self Time | Total Time | CTC. Function | Convergence Rate |
---|---|---|---|---|

11 ´ 11 | 0.312 s | 0.743 s | 12,642 | 2.8201 |

31 ´ 31 | 1.094 s | 2.291 s | 38,322 | 2.8934 |

51 ´ 51 | 1.968 s | 4.030 s | 65,602 | 2.8733 |

Grid Spacing | Self Time | Total Time | CTC. Function | Convergence Rate |
---|---|---|---|---|

h | 12.757 s | 18.458 s | 293,250 | 2.8201 |

h/2 | 3.18925 s | 4.6145 s | 73,356 | 2.8934 |

In this research article, Crank Nicolson scheme has been successfully applied to find the solutions of two-dimension nonlinear reaction diffusion system. The accuracy and stability of the scheme demonstrated by test problem with data tables and figures. According to T Lakoba [

computational time and increase memory capacity [

is that the implicit solver only requires a tridiagonal matrix algorithm to be solved, so that the difference between the true Crank Nicolson solution and ADI approximated solution has an order of accuracy of O ( k 2 ) and hence can be ignored with a sufficiently small time-step [

performance of the CPU for two different schemes. The derivation of our ADI scheme for a nonlinear PDE system relies on a few key observations. Most importantly, using the solution at time levels previous to t = t n + 1 , the algorithm converts the nonlinear spatial operator into an implicit but linear operator with variable coefficients. The resulting approximately-factored equation is solved in sweeps along each of the Cartesian directions, including, as is common in ADI approaches, an intermediate t n + 1 / 2 step, so that all of the proposed algorithms are embodied in the two steps formula that every iteration updated the block tridiagonal linear algebraic system [

Since last two decades, there is a challenging hasting among vendors and software development communities to bring improvement in performance for

High performance computing system, by halting the traditional development to increase the clock rate, number of cores that are being increased in the system, however, many cores architecture based different powerful devices such GPU (Graphics Processing Unit) and GPGPU (General Purpose Graphics Processing Unit) by NIVIDA, MIC (Many Integrated Core) by Intel and FPGA (Field programmable Gate Array) have been introduced recently that outperform the conventional CPU processing by thousand folds [

Keeping in view, the advantages of this emerging technology, we have introduced Crank Nicolson and ADI schemes with the help of this application by using FUJITSU Primergy RX 350 S7 HPC computer having Intel Xeon E5-2667 processor of 2.80 GHz processing power which contained 16 physical cores and 32 logical cores, main memory size of 32 GB and HDD 4 TB inside it [

The authors are gratefully acknowledged Dr. Muhammad Faheem Afzaal, Department of Chemical Engineering, Imperial College London, UK and Muhammad Usman Ashraf, Department of Computer Science, King Abdulaziz University, Saudi Arabia.

The authors do not have any conflict of interest in this research paper.

Hasnain, S., Saqib, M. and Al-Harbi, N. (2018) Finite Difference Implicit Schemes to Coupled Two-Dimension Reaction Diffusion System. Journal of Applied Mathematics and Physics, 6, 737-753. https://doi.org/10.4236/jamp.2018.64066