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To develop an efficient numerical scheme for two-dimensional convection diffusion equation using Crank-Nicholson and ADI, time-dependent nonlinear system is discussed. These schemes are of second order accurate in apace and time solved at each time level. The procedure was combined with Iterative methods to solve non-linear systems. Efficiency and accuracy are studied in term of
L_{2}
,
L<sub>_{}
∞</sub>
norms confirmed by numerical results by choosing two test examples. Numerical results show that proposed alternating direction implicit scheme was very efficient and reliable for solving two dimensional nonlinear convection diffusion equation. The proposed methods can be implemented for solving non-linear problems arising in engineering and physics.

In this paper we have extended our previous approach associated to two dimension Convection-diffusion equation. The great Physicist Johannes Martinus Burgers discovered Burgers equation, which is non-linear parabolic partial dif- ferential equation (PDE) and widely used as a model in many engineering problems, which explains such as physical flow phenomena in fluid dynamics, turbulence, boundary layer behavior, shock wave formation, and mass transport [

u t + u u x + u u y − 1 R ( u x x + u y y ) = 0 (1)

where ( x , y , t ) ∈ Ω × ( 0 , T ]

with initial conditions

u ( x , y , 0 ) = u 0 ( x , y ) , ( x , y ) ∈ Ω

The Dirichlet boundary conditions are given by

u ( a , y , t ) = f 1 ( x , y , t ) , u ( b , y , t ) = f 2 ( x , y , t )

u ( x , c , t ) = g 1 ( x , y , t ) , u ( x , d , t ) = g 2 ( x , y , t )

where ( x , y , t ) ∈ Ω × ( 0 , T ] , Ω = { ( x , y ) : a ≤ x ≤ b , c ≤ y ≤ d } is a rectangular domain in R 2 , ( 0 , T ] is the time interval. u 0 , f 1 , f 2 , g 1 , g 2 are given sufficiently smooth functions and u ( x , y , t ) may represent heat, diffusion, etc. Re is the Reynolds number.

This equation established the interaction between the non-linear convection processes and the diffusive viscous processes [

Many different researchers used Burgers equation to develop new algorithms and to test various existing algorithms [

From literature review, [

u ( x , y , t ) = 0.5 − tanh ( ( x + y − t ) R 2 ) (2)

where ( x , y ) ∈ Ω , t > 0 and R is a parameter, known as Reynolds number. Boundary conditions and initial conditions can be taken from exact solution of u(x,y,t) [

In this problem the rectangular domain of two dimensional nonlinear convection- diffusion Equation (1) is given as Ω = [ ( x , y ) : 0 ≤ x ≤ 1,0 ≤ y ≤ 1 ] . Exact solution of the above two dimensional equation is

u ( x , y , t ) = 1 1 + ( x + y − t ) R 2 (3)

where ( x , y ) ∈ Ω , t > 0 and R is a parameter, known as Reynolds number. Boundary conditions and initial conditions can be taken from exact solution of u ( x , y , t ) . where Ω is a rectangular domain in R 2 . The main objective of the paper is to find efficient solution of unknown u ( x , t ) . Two test problems were described to understand the numerical solution by taking two finite difference schemes. Also Convection diffusion equation has been extensively studied to describe various kinds of phenomena which can be seen from equation [

Numerical solution of the two dimensional non-linear equation in a finite domain Ω . The first step is to choose integers L and M to define step sizes h x = ( b − a ) / L and h y = ( d − c ) / M in x and y directions respectively. Partition the interval [a, b] into L equal parts of width h x and the interval [c, d] into M equal parts of width h y . Place a grid by drawing vertical and horizontal lines through the points with coordinates ( x l , y m ) , where x l = a + l h for each l = 0 , 1 , 2 , ⋯ , L and y m = c + m k for each m = 0 , 1 , 2 , ⋯ , M also the lines x = x l and y = y m are grid lines, and their intersections are the mesh points of the grid. For each mesh point in the interior of the grid, ( x l , y m ) , for l = 1 , 2 , ⋯ , L − 1 and m = 1 , 2 , ⋯ , M − 1 , we apply different algorithms to approximate the numerical solution to the problem in equation [

We apply Crank-Nicholson implicit finite difference scheme to equation [

u t = u l , m n + 1 + u l , m n k , u = u l , m n + 1 + u l , m n 2 u x = u l + 1 , m n + 1 − u l − 1 , m n + 1 + u l + 1 , m n − u l − 1 , m n 4 h , u y = u l , m + 1 n + 1 − u l , m − 1 n + 1 + u l , m + 1 n − u l , m − 1 n 4 h 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 }

when substitute these terms in to Equation (1), the Crank-Nicholson Scheme is given by

u l , m n + 1 − u l , m n + R 1 [ ( u l , m n + 1 + u l , m n ) { u l + 1 , m n + 1 − u l − 1 , m n + 1 + u l + 1 , m n − u l − 1 , m n + u l , m + 1 n + 1 − u l , m − 1 n + 1 + u l , m + 1 n − u l , m − 1 n } ] + R 2 [ u l + 1 , m n + 1 + u l − 1 , m n + 1 + u l , m + 1 n + 1 + u l , m − 1 n + 1 − 4 u l , m n + 1 + u l + 1 , m n + u l − 1 , m n + u l , m + 1 n + u l , m − 1 n − 4 u l , m n ] = 0 where R 1 = k 8 h , R 2 = k 2 R h 2 (4)

The scheme shows that the accuracy is of O ( k 2 + h 2 ) . A Jacobian matrix is now Penta-diagonal, but unfortunately due to large number of iterations it extends from the diagonal at least n entries away in every direction,but another methods which can be used to handle such problems (discussed later), because of the large bandwidth, increasing grid points the calculation become more difficult. To overcome this difficulty another method solution is needed.Newton method is used for solving nonlinear task (discussed later). The Crank- Nicholson is computationally inefficient.

In search of a time efficient alternate, we analyzed that the Crank-Nicholson scheme for the two dimensional equation, and find out that scheme is not time efficient [

In this approach, the finite difference equations are written in terms of quantities at two levels However, two different finite difference approximations were used alternately, one to advance the calculations from the plane n to a plane n*, and the second to advance the calculations from (n*)-plane to the (n + 1). Same parameters were used in this method as described above. The derivation of ADI scheme, we have following steps;

Sweep in x-direction

u l , m * − u l , m n = P 1 ( u l + 1 , m * − 2 u l , m * + u l − 1 , m * + u l + 1 , m n − 2 u l , m n + u l − 1 , m n ) + P 2 ( u l , m + 1 n − 2 u l , m n + u l − 1 , m n ) + P 3 ( u l , m n ( u l + 1 , m n − u l − 1 , m n ) ) + P 3 ( u l , m n ( u l , m + 1 n − u l , m − 1 n ) ) (5)

u l , m * − u l , m n = P 1 ( δ x 2 ( u l , m * + u l , m n ) ) + P 2 ( δ y 2 ( u l , m n ) ) + k F ( u n ) (6)

where

F ( u n ) = P 3 ( u l , m n δ x ( u l , m n ) ) + P 3 ( u l , m n δ y u l , m n )

Sweep in y-direction

u l , m n + 1 − u l , m * = P 1 ( u l , m + 1 n + 1 − 2 u l , m n + 1 + u l , m − 1 n + 1 + u l , m + 1 * − 2 u l , m * + u l , m − 1 * ) + P 2 ( u l + 1 , m * − 2 u l , m * + u l − 1 , m * ) + P 3 ( u l , m * ( u l + 1 , m * − u l − 1 , m * ) ) + P 3 ( u l , m * ( u l , m + 1 * − u l , m − 1 * ) ) (7)

u l , m n + 1 − u l , m * = P 1 ( δ y 2 ( u l , m n + 1 + u l , m * ) ) + P 2 ( δ x 2 ( u l , m * ) ) + k F ( u * ) (8)

where

F ( u * ) = P 3 ( u l , m * δ x ( u l , m * ) ) + P 3 ( u l , m * δ y u l , m * )

P 1 = k R e 2 h 2 , P 2 = k R e h 2 , P 3 = k 2 h

where u l , m * defines similarly to u l , m n + 1 . This method is unconditionally stable. The method has accuracy O ( k 2 + h 2 ) , newton’s iterative method is used to solve tridiagonal system.

The family of linear system in x-direction as:

a 1 u x , m u * = b 1 m + F ( k u x , m n ) (9)

where m = 1 , ⋯ , M − 1

The family of linear system in y-direction as:

c 1 u y , m n + 1 = d 1 m + F ( k u y , m * ) (10)

where l = 1 , ⋯ , L − 1

where a 1 , c 1 develops tridiagonal matrix and the array b 1 , d 1 depends on l and m

The reaction term is x-direction

F ( k u x , m n ) = [ k F ( u 1, m n ) , ⋯ , k F ( u 1, L − 1 n ) ]

similarly for the reaction term in y-direction:

F ( k u y , l * ) = [ k F ( u 1 , M − 1 * ) , ⋯ , k F ( u 1 , 1 * ) ]

finally the scheme makes tridiagonal family of linear system.Iterative methods was carried out to solved this system. The trick used in constructing the ADI scheme is to split time step into two, 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 are chosen such that the resulting linear system is tridiagonal [

Algorithm 1

To construct Newton iterative method for the two dimensional Convection- diffusion equation. The non-linear system in equations [

G ( S ) = 0 (11)

where

S _ ≈ [ u _ n + 1 ] = [ S 1, m , S 2, m , ⋯ , S ( 2 L − 2 ) , m ] T

u _ : m n + 1 = [ u 1, m n + 1 , u 2, m n + 1 , u 3, m n + 1 , ⋯ , u L − 1, m n + 1 ] T

G _ = [ G 1, m , G 2, m , ⋯ , G ( 2 L − 2 ) , m ] T , where G 1, m , G 2, m , ⋯ , G ( 2 L − 2 ) , m were system of nonlinear equations obtained from the system in [

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

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

・ Solve the linear system A ( u ( k ) ) Δ u ( k ) = − R ( u ( k ) )

・ Specify u ( k + 1 ) = u ( k ) + Δ u ( k ) ,

where A ( u ( k ) ) is ( m × m ) Jacobian matrix, which is computed analytically and Δ u ( k ) is the correction vector. In the iteration method solution at the previous time step is taken as the initial guess. Iteration at each time step is stopped when ‖ R ( u ( k ) ) ‖ ∞ ≤ T o l with Tol is a very small prescribed value. The linear system obtained from Newton’s iterative method, is solved by Court’s method. Convergence done with iterations along less CPU time [

Algorithm 2

Clearly, the system is tridiagonal and can be solved with Thomas algorithm. The dimension of J is l × m . In general a tridiagonal system can be written as,

a l x l − 1 + b l x l + c l x l + 1 = S l , with a 1 = c l = 0

above system can be written as in a matrix-vector form,

J u = S

where J is a coefficient matrix (Jacobean Matrix), which is known, comes from Newton’s iterative method. Right hand side is column vector which is known.Our main goal is to find the resultant vector u . Now we have

J = [ b 1 c 1 0 0 0 ⋯ 0 a 2 b 2 c 2 0 0 ⋯ 0 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ a n − 1 b n − 1 c n − 1 ⋯ ⋯ ⋯ ⋯ a n b n ]

u _ = [ u 1 , u 2 , u 3 , ⋯ , u n ] t

S _ = [ s 1 , s 2 , s 3 , ⋯ , s n ] t

technique is explained in the following steps,

J = L U

where

L = [ μ 1 0 0 0 0 ⋯ 0 β 2 μ 2 0 0 0 ⋯ 0 α 3 β 3 μ 3 0 0 ⋯ 0 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ 0 ⋯ ⋯ α n − 1 β n − 1 μ n − 1 0 ⋯ ⋯ ⋯ α n β n μ n ]

and

U = [ 1 δ 1 λ 1 0 0 ⋯ 0 0 1 δ 2 λ 2 0 ⋯ 0 0 0 1 δ 3 λ 3 ⋯ 0 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ 0 ⋯ ... 0 1 δ n − 2 λ 2 ⋯ ⋯ 0 0 1 δ n − 1 ⋯ ⋯ ⋯ 0 0 1 ]

By equating both sides of the J u = S , we get the elements of the matrices L and U . The computational tricks for the implementation of Thomas algorithm are shown in results, taken from a specific examples.

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

R M S E r r o r = ∑ i , j = 1 L ( U i , j − u i , j ) 2 L × L (12)

L ∞ = max 1 ≤ i ≤ L ∑ j = 1 L | ( U i . j − u i , j ) | (13)

L 2 = ρ ( U i , j − u i , j ) t ( U i , j − u i , j ) (14)

where u ( x , y , t ) and U ( x , y , t ) denote the numerical and exact solutions at the grid point ( x l , y m , t n ) . In this method ρ ( U i , j − u i , j ) = max ( λ ) and λ is an eigen value of ( U i , j − u i , j ) respectively.

Numerical computations were performed using the uniform grid. In

In problem 2, considering Equation (1) over the domain [0.5, 0.5] × [0.5, 0.5], boundary and initial conditions were taken from the exact solution showed stable results time step and increasing grid size (refine mesh size). In

In

Solution Comparison | |||
---|---|---|---|

(Typical mesh points) | CN | ADI | Exact |

(−0.3, −0.3) | −0.5000 | −0.5000 | −0.5000 |

(−0.45, −0.45) | −0.49999 | −0.5000 | −0.5000 |

(−0.3, −0.45) | −0.49999 | −0.49999 | −0.5000 |

(−0.35, −0.35) | −0.49998 | −0.49997 | −0.5000 |

(−0.25, 0.25) | 0.0000 | −0.0000 | 0.0000 |

(−0.2, −0.2) | −0.49997 | −0.49888 | −0.5000 |

(−0.2, −0.3) | 0.49997 | 0.49888 | 0.5000 |

(0.2, 0.2) | 0.49998 | 0.49998 | 0.5000 |

(0.2, 0.35) | −0.49999 | −0.49999 | −0.5000 |

(0.3, 0.2) | 0.49999 | 0.49777 | 0.5000 |

(0.25, −0.25) | 0.0000 | 0.0000 | 0.0000 |

(0.4, 0.4) | 0.5000 | 0.5000 | 0.5000 |

Solution Comparison | |||
---|---|---|---|

(Typical mesh points) | CN | ADI | Exact |

(−0.3, −0.3) | −0.50000 | −0.50000 | −0.5000 |

(−0.45, −0.45) | −0.48999 | −0.50000 | −0.5000 |

(−0.3, −0.45) | −0.48999 | −0.489999 | −0.5000 |

(−0.35, −0.35) | −0.48998 | −0.489997 | −0.5000 |

(−0.25, 0.25) | 0.00000 | −0.00000 | 0.0000 |

(−0.2, −0.2) | −0.489997 | −0.489888 | −0.5000 |

(−0.2, −0.3) | 0.489997 | 0.489888 | 0.5000 |

(0.2, 0.2) | 0.489998 | 0.489998 | 0.5000 |

(0.2, 0.35) | −0.489999 | −0.489999 | −0.5000 |

(0.3, 0.2) | 0.489999 | 0.489777 | 0.5000 |

(0.25, −0.25) | 0.00000 | 0.00000 | 0.0000 |

(0.4, 0.4) | 0.50000 | 0.50000 | 0.5000 |

Solution Comparison | |||
---|---|---|---|

(Typical mesh points) | CN | ADI | Exact |

(−0.3, −0.3) | −0.50000 | −0.50000 | −0.5000 |

(−0.45, −0.45) | −0.49999 | −0.50000 | −0.5000 |

(−0.3, −0.45) | −0.49999 | −0.489999 | −0.5000 |

(−0.35, −0.35) | −0.49998 | −0.489997 | −0.5000 |

(−0.25, 0.25) | −0.49997 | −0.49997 | −0.5000 |

(−0.2, −0.2) | −0.499997 | −0.489888 | −0.5000 |

(−0.2, −0.3) | −0.499997 | −0.489888 | −0.5000 |

(0.2, 0.2) | −0.499998 | −0.489998 | −0.5000 |

(0.2, 0.35) | −0.499999 | −0.489999 | −0.5000 |

(0.3, 0.2) | −0.499999 | −0.489777 | −0.5000 |

(0.25, −0.25) | −0.49999 | −0.49999 | −0.5000 |

(0.4, 0.4) | −0.50000 | −0.50000 | −0.5000 |

Solution Comparison | |||
---|---|---|---|

(Typical mesh points) | CN | ADI | Exact |

(−0.3, −0.3) | −0.50000 | −0.50000 | −0.5000 |

(−0.45, −0.45) | −0.4998 | −0.50000 | −0.5000 |

(−0.3, −0.45) | −0.4997 | −0.4899 | −0.5000 |

(−0.35, −0.35) | −0.4887 | −0.4897 | −0.5000 |

(−0.25, 0.25) | −0.4886 | −0.4987 | −0.5000 |

(−0.2, −0.2) | −0.4886 | −0.4787 | −0.5000 |

(−0.2, −0.3) | −0.4998 | −0.4997 | −0.4998 |

(0.2, 0.2) | −0.5000 | −0.4999 | −0.5000 |

(0.2, 0.35) | −0.4546 | −0.4447 | −0.4847 |

(0.3, 0.2) | −0.4999 | −0.4999 | −0.5000 |

(0.25, −0.25) | −0.4889 | −0.4776 | −0.4998 |

(0.4, 0.4) | −0.4998 | −0.5998 | −0.4998 |

Solution Comparison | |||
---|---|---|---|

(Typical mesh points) | CN | ADI | Exact |

(0.2, 0.2) | 1.0000 | 1.0000 | 1.0000 |

(0.05, 0.05) | 0.9996 | 0.9887 | 1.0000 |

(0.2, 0.05) | 0.9765 | 0.9665 | 1.0000 |

(0.15, 0.15) | 0.8514 | 0.8661 | 1.0000 |

(0.3, 0.3) | 0.8515 | 0.8415 | 1.0000 |

(0.3, 0.2) | 0.9000 | 0.9112 | 1.0000 |

(0.7, 0.7) | 0.7515 | 0.7414 | 1.0000 |

(0.7, 15) | 0.8775 | 0.8675 | 0.9987 |

(0.8, 0.3) | 0.9876 | 0.9866 | 1.0000 |

(0.9, 0.9) | 0.9996 | 0.9954 | 1.0000 |

(0.25, 0.75) | 0.0000 | 0.0000 | 0.0000 |

(0.75, −0.25) | 1.0000 | 1.0000 | 1.0000 |

Solution Comparison | |||
---|---|---|---|

(Typical mesh points) | CN | ADI | Exact |

(0.2, 0.2) | 1.0000 | 1.0000 | 1.0000 |

(0.05, 0.05) | 1.0000 | 1.0000 | 1.0000 |

(0.2, 0.05) | 1.0000 | 1.0000 | 1.0000 |

(0.15, 0.15) | 0.9999 | 0.9998 | 1.0000 |

(0.3, 0.3) | 0.9999 | 0.9998 | 1.0000 |

(0.3, 0.2) | 0.9997 | 0.99996 | 1.0000 |

(0.7, 0.7) | 1.0000 | 1.0000 | 1.0000 |

(0.7, 15) | 0.9999 | 0.9998 | 1.0000 |

(0.8, 0.3) | 0.9889 | 0.9888 | 1.0000 |

(0.9, 0.9) | 1.0000 | 1.0000 | 1.0000 |

(0.25, 0.75) | 0.9888 | 0.9887 | 1.0000 |

(0.75, −0.25) | 0.9888 | 0.9887 | 1.0000 |

k | grid size | Mittal [ | Mittal [ | Khater [ | Khater [ | |||
---|---|---|---|---|---|---|---|---|

0.005 | 4.9586e−007 | 5.9234e−007 | 4.9688e−008 | 4.6512e−008 | 1.19e−007 | 8.94e−008 | ||

0.0005 | 6.1723e−007 | 6.6123e−007 | 6.2171e−009 | 5.9068e−009 | 8.05e−007 | 7.45e−007 | ||

0.0001 | 2.1132e−008 | 3.1324e−008 | 2.5318e−009 | 2.188e−009 | .... | .... | ||

0.0001 | 1.2654e−008 | 2.2131e−008 | 1.8750e−009 | 1.0101e−009 | ... | ... | ||

k | grid size | Mittal [ | Mittal [ | Khater [ | Khater [ | ||
---|---|---|---|---|---|---|---|

0.005 | 9.9586e−008 | 9.734e−008 | 9.9899e−009 | 9.8076e−009 | 1.70e−007 | 1.50e−007 | |

0.0005 | 7.4323e−008 | 7.0323e−008 | 8.1271e−009 | 7.0450e−009 | 9.82e−007 | 8.50e−007 | |

0.0001 | 6.1132e−008 | 6.0724e−008 | 7.2310e−009 | 6.0548e−009 | .... | .... | |

0.0001 | 3.2554e−008 | 3.0231e−008 | 3.9366e−009 | 3.0601e−009 | ... | ... |

grid size | Mittal [ | Mittal [ | Rms | ||
---|---|---|---|---|---|

4.5112e−016 | 3.4661e−016 | 5.4017e−017 | 4.5997e−017 | 4.5112e−016 | |

2.3451e−016 | 1.5712e−016 | 3.6338−017 | 2.6712e−017 | 2.3451e−016 | |

1.0231e−017 | 1.0321e−017 | 1.2898e−017 | 1.1275e−017 | 1.0231e−017 |

grid size | Mittal [ | Mittal [ | Rms | ||
---|---|---|---|---|---|

4.41172e−005 | 3.8761e−005 | 5.0688e−006 | 4.9850e−006 | 4.41172e−005 | |

2.3151e−005 | 2.9712e−005 | 3.4403e−006 | 3.2143e−006 | 2.3151e−005 | |

1.0931e−006 | 1.0991e−006 | 2.3549e−006 | 2.0547e−006 | 1.0931e−006 |

size and grid size with fixed Reynolds no Re = 1, t = 0.05 and 0.25. Good results obtained when compared the values of this exact solution with those of the approximation gained in Tables 1-3. Furthermore in

Results gained by using ADI scheme at very small step spacing to understand the importance of reducing steps. Sharp edges remove during increases time level. These results are very interesting for us to understand the efficiency of the ADI scheme. The corresponding graphical representation for the solution of unknown u ( x , y , t ) was presented in

Re | grid size | k | Mittal [ | Mittal [ | Rms | ||
---|---|---|---|---|---|---|---|

100 | 0.001 | 6.1601e−005 | 5.9645e−005 | 7.1536e−006 | 6.9729e−006 | 6.1601e−005 | |

300 | 0.0001 | 7.2134e−004 | 7.1206e−004 | 8.2579e−005 | 8.1304e−005 | 7.2134e−004 | |

500 | 0.0001 | 2.6145e−004 | 1.4172e−004 | 3.6692e−004 | 2.4573e−004 | 2.6145e−004 |

have been obtained in earlier studies (Mittal and Jiwari, 2016). In literature point of view present schemes shows similar results as (Jain and Holla, 1978; Mittal and Jiwari, 2016, khater (2008)) [

The obtained results gives excellent agreement with the solutions available in the literature. When the number of grid points get larger than several hundred, the memory and storage of the Crank-Nicholson starts to become a serious issue and it is better to solve this method using different approaches that take advantage of the special form. This method is computationally inefficient. Thomas algorithm avoids having to store having the whole matrix J (Jacobean) in its memory and solve the system much more expediently. ADI methods reduces to the CN scheme and this method solve the system very efficiently.The order of truncation error: O ( k 2 + h 2 ) . The implementation of ADI computa- tionally is in a time efficient manner. Alternating direction implicit method is fastest when it works and it works well for simple,ideal problems and give efficient results.

In this research work, finite difference methods has been discussed for solving two dimensional convection-diffusion equation. Two test problems were considered, explained the efficiency, accuracy and stability of the schemes. The numerical results showed that Alternating Direction Implicit method is easy to implement and excellent in time efficient manner. The accuracy and stability of these methods were compared to the other numerical methods, shows good agreement with the exact solution. Both ADI and Crank-Nicholson are un- conditionally stable and highly accurate. For convergence L 2 and L ∞ norms were treated towards zero when grid size was increased. Numerical results showed that both methods are good but ADI method is consistent and time- efficient. The approach used in this paper may be useful to solve higher dimensional partial differential equations appearing in various applications of science and engineering.

This research was supported by the division of Numerical Analysis, department of Mathematics, King Abdulaziz University. Authors are thankful to Vineet K. Srivastava Scientist/Engineer-ISTRAC/ISRO, Bangalore, India and M.S. Ismail Department of Maths, King Abdulaziz University, Jeddah, Kingdom of Saudia Arabia, for providing help to understand the numerical technique for highly accurate solution.

This paper has no financial or non-financial competing interest.

The authors contributed to the manuscripts equally.

Saqib, M., Hasnain, S. and Mashat, D.S. (2017) Computational Solutions of Two Dimensional Convection Diffusion Equation Using Crank- Nicolson and Time Efficient ADI. American Journal of Computational Mathematics, 7, 208-227. https://doi.org/10.4236/ajcm.2017.73019