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Stability of Two Dimensional FTSC scheme: By Fourier stability analysis the equation (6.4.36) can be written as ................. (6.4.39) The error equation is ................. (6.4.40) Making use of (6.4.29), equation (6.4.40) can be written as The condition needed for stability is . Therefore since r₁, r₂, are all not negative quanities, the only condition to be satisfied here is ................. (6.4.41) since the maxumum values of is one, the condition for the FTCS scheme to two dimensional diffusion equation to be stable is . Crank-Nicolson scheme to Two-Dimensional diffusion equation: Consider the average of FTCS scheme (6.4.40) and the fully implicit scheme ................. (6.4.42) to two dimensional heat equation (6.4.35); ................. (6.4.43) Separating (n+1) ^th time level terms to left hand side of the equation and the known n ^th time level values to the right hand side of the equation gives ................. (6.4.44) for i = 1, 2 . . . N-1, j = 1, 2, . . . M-1 and n = 0, 1, 2 . . . where . Equation (6.4.44) in matrix form can be written as ................. (6.4.45) where is the unknown, the known vectors and the coefficient matrix is a square matrix of order s where *s=(N-1)(M-1). The structure of is In the matrix the position of the term from the principal diagonal element in horizontal direction in any row or in vertical direction in any column is equal to the number of unknown elements in x direction if the elements in the unknown vector are written by taking elements in the direction of x first and then in the direction of y. This value is equal to the unknown elements in y - direction if the elements in the unknown vector are written by taking elements in the direction of y first and then in the direction of x. These elements are the coefficients of the terms on the left hand side of (6.4.44). The coefficient matrix in (6.4.44) is also a sparse matrix, however, in (6.4.44) there are two extra non-zero diagonals at a distance of (N-1) from the principal diagonal element in each row. Thomas algorithm which has been used to solve the system(6.4.7) obtained by Crank-Nicolson scheme to one-dimensional equation cannot used to solve (6.4.44) because of these extra non-zero diagonals. An alternative is to use the full Gaussian elimination procedure but unfortunately this method initially fills some of the zero elements of the coefficient matrix and then make them zero again. That is, using Gaussian elimination to solve the system (6.4.44) is not an economical process and also the scheme requires to store the huge matrix during the soluion process. Alternate direction implicit (ADI) method to two dimensional diffusion equations. This method is also similar to fully implicit scheme implemented in two steps. In the first step the implicit terms (n+1 th time level terms) on the right hand side of (6.4.42) will be taken only in one direction of x and y at half the step length in time direction (that is at n+1/2) and in the second step the implicit terms will be taken in the other space direction at n+1 th time level. With this modification equation (6.4.42) can be written as ................. (6.4.46) ................. (6.4.47)** for i = 1, 2, 3 . . . N-1, j = 1, 2, 3 . . . M-1, and n = 0,1,2 . . . Equation (6.4.46) can be written as ................. (6.4.48) This is a tri diagonal system which can be solved using Thomas algorithm for the unknown u_{i, j}at the time level n+1/2. Similarly equation (6.4.54b) can be written as ................. (6.4.49) since u terms on the right hand side of (6.4.47b) are already been calculated by solving (6.4.47a), (6.4.47b) is again a tri-diagonal system which can also be solved using Thomas algorithm for u_{i, j} at time level n+1. This completes one iteration in time direction and the same is repeated until the desired time level is reached.

Stability of Two Dimensional FTSC scheme:

By Fourier stability analysis the equation (6.4.36) can be written as

................. (6.4.39)

The error equation is

................. (6.4.40)

Making use of (6.4.29), equation (6.4.40) can be written as

The condition needed for stability is .

Therefore

since r₁, r₂, are all not negative quanities, the only condition to be satisfied here is

................. (6.4.41)

since the maxumum values of is one, the condition for the FTCS scheme to two dimensional diffusion equation to be stable is .

Crank-Nicolson scheme to Two-Dimensional diffusion equation:

Consider the average of FTCS scheme (6.4.40) and the fully implicit scheme

................. (6.4.42)

to two dimensional heat equation (6.4.35);

................. (6.4.43)

Separating (n+1) ^th time level terms to left hand side of the equation and the known n ^th time level values to the right hand side of the equation gives

................. (6.4.44)

for i = 1, 2 . . . N-1, j = 1, 2, . . . M-1 and n = 0, 1, 2 . . .

where .

Equation (6.4.44) in matrix form can be written as

................. (6.4.45)

where is the unknown, the known vectors and the coefficient matrix is a square matrix of order s where s=(N-1)*(M-1). The structure of is

In the matrix the position of the term from the principal diagonal element in horizontal direction in any row or in vertical direction in any column is equal to the number of unknown elements in x direction if the elements in the unknown vector are written by taking elements in the direction of x first and then in the direction of y. This value is equal to the unknown elements in y - direction if the elements in the unknown vector are written by taking elements in the direction of y first and then in the direction of x. These elements are the coefficients of the terms on the left hand side of (6.4.44).

The coefficient matrix in (6.4.44) is also a sparse matrix, however, in (6.4.44) there are two extra non-zero diagonals at a distance of (N-1) from the principal diagonal element in each row. Thomas algorithm which has been used to solve the system(6.4.7) obtained by Crank-Nicolson scheme to one-dimensional equation cannot used to solve (6.4.44) because of these extra non-zero diagonals. An alternative is to use the full Gaussian elimination procedure but unfortunately this method initially fills some of the zero elements of the coefficient matrix and then make them zero again. That is, using Gaussian elimination to solve the system (6.4.44) is not an economical process and also the scheme requires to store the huge matrix during the soluion process.

Alternate direction implicit (ADI) method to two dimensional diffusion equations.

This method is also similar to fully implicit scheme implemented in two steps.

In the first step the implicit terms (n+1 th time level terms) on the right hand side of (6.4.42) will be taken only in one direction of x and y at half the step length in time direction (that is at n+1/2) and in the second step the implicit terms will be taken in the other space direction at n+1 th time level. With this modification equation (6.4.42) can be written as

................. (6.4.46)

................. (6.4.47)

for i = 1, 2, 3 . . . N-1, j = 1, 2, 3 . . . M-1, and n = 0,1,2 . . .

Equation (6.4.46) can be written as

................. (6.4.48)

This is a tri diagonal system which can be solved using Thomas algorithm for the unknown u_{i, j}at the time level n+1/2. Similarly equation (6.4.54b) can be written as

................. (6.4.49)

since u terms on the right hand side of (6.4.47b) are already been calculated by solving (6.4.47a), (6.4.47b) is again a tri-diagonal system which can also be solved using Thomas algorithm for u_{i, j} at time level n+1. This completes one iteration in time direction and the same is repeated until the desired time level is reached.