Untitled Document

Stability of the numerical schemes In application of any numerical scheme to solve a differential equation, if small change in the input data leads to only a small change in the output data then such a numerical scheme is called stable. Since numerical solutions obtained using computer are proned to round off errors, unless the numerical scheme under use is stable, these errors will get magnified during the iterative process and even dominate the solution. There are two different approaches to study the stability of the numerical schemes. In the first method, the difference equations are expressed in matrix form and the eigen values of the associated matrix are examined. In the other, finite fourier series is used to analyze the stability. The later method is much simpler to use than the former however, the later method is less rigorous because it neglects the boundary conditions. Matrix method of Stability of FTCS scheme: Consider the explicit FTCS scheme (6.4.2) to one dimensional diffusion equation; for i = 1, 2, 3 . . . N-1 and n = 0, 1, 2 . . . with boundary conditions u = 0 at x = 0 and 1 for all t greater than zero and with some known initial values of u at t=0. The system of equations (6.4.2) for a particular time level say 'n', in matrix form can be written as ................. (6.4.15) where and is the *(N-1)(N-1) triangular matrix of the form Equation (6.4.15), can be rewriten as ................. (6.4.16) where the vector is u with some known initial values at time t=0. Since any numerical solution is bound to rounding errrors, let us assume that at time t=0 computations are started with instead of . That is, So that, the total error at n^th time level is ................. (6.4.17) Equation (6.4.17) is similar to (6.4.16) except e in the place of u shows that the formula for the propagation of the error is same as that for the solution of u. Assume that the matrix is non-deficient, that is, it has (N-1) linearly independent eigen vectors (which is always the case if has distinct eigen values l_i or the coefficient matrix is real and symmetric). Then these (N-1) linearly independent eigen vectors can form a basis for the (N-1) dimensional space in which is one of the elements. Hence, ................. (6.4.18) where c_i, i = 1, 2, 3 . . . N-1 are some known constants. Now consider the equation (6.4.17) to find the error at the n ^th time level ................. (6.4.19) This error has to be bounded as the number of iterations 'n' increases infinitely, that is as n--->0. From equation (6.4.19) this condition is satisfied whenever maximum of \|l_i\| < 1 that is the maximum of the absolute value of the eigen values of is less than one. So to check for the stability of any numerical scheme it is sufficient to check for the modulus of the largest eigen value of the coefficient matrix is to be less than unity. For FTCS scheme the coefficient matrix is ................. (6.4.20) where I is the identity matrix of order (N-1)(N-1)* and T_N-1 is a square matrix of order (N-1) with eigen values ................. (6.4.21) and the corresponding eigen vectors are ................. (6.4.22) From (6.4.20) and (6.4.21) the eigen values of are ................. (6.4.23) Hence the condition for stability is Since is always positive, the only useful condition is ................. (6.4.24) Conditon (6.4.24) is always true whenever that is, the condition for FTCS scheme is to be stable is r must be less or equal to 1/2. This is precisely what has been observed in the example 6.4.1. For this problem if FTCS scheme with r > .5 is used, to obtain the numerical solution the solution became unphysical. Stability of the Crank-Nicolson scheme: Consider the Crank-Nicolson scheme to the one-dimensional diffusion equation with zero boundary conditions and some known initial conditions(6.4.7) for i = 1, 2,.......N-1 and r = 0, 1, 2................. Therefore the iterative matrix for the present scheme is and the eigen values of this matrix are ................. (6.4.25) In(6.4.25) for all positive values of r the denominator is greater than the numerator makes modulus of eigen values of less than one. Hence Crank-Nicolson scheme is unconditionally stable. This is again reflected in the example 6.4.2 because the scheme was giving reasonable results even with r=1.

Stability of the numerical schemes

In application of any numerical scheme to solve a differential equation, if small change in the input data leads to only a small change in the output data then such a numerical scheme is called stable. Since numerical solutions obtained using computer are proned to round off errors, unless the numerical scheme under use is stable, these errors will get magnified during the iterative process and even dominate the solution.

There are two different approaches to study the stability of the numerical schemes. In the first method, the difference equations are expressed in matrix form and the eigen values of the associated matrix are examined. In the other, finite fourier series is used to analyze the stability. The later method is much simpler to use than the former however, the later method is less rigorous because it neglects the boundary conditions.

Matrix method of Stability of FTCS scheme:

Consider the explicit FTCS scheme (6.4.2) to one dimensional diffusion equation;

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

with boundary conditions u = 0 at x = 0 and 1 for all t greater than zero and with some known initial values of u at t=0.

The system of equations (6.4.2) for a particular time level say 'n', in matrix form can be written as

................. (6.4.15)

where

and is the (N-1)*(N-1) triangular matrix of the form

Equation (6.4.15), can be rewriten as

................. (6.4.16)

where the vector is u with some known initial values at time t=0.

Since any numerical solution is bound to rounding errrors, let us assume that at time t=0 computations are started with instead of . That is,

So that, the total error at n^th time level is

................. (6.4.17)

Equation (6.4.17) is similar to (6.4.16) except e in the place of u shows that the formula for the propagation of the error is same as that for the solution of u.

Assume that the matrix is non-deficient, that is, it has (N-1) linearly independent eigen vectors (which is always the case if has distinct eigen values l_i or the coefficient matrix is real and symmetric). Then these (N-1) linearly independent eigen vectors can form a basis for the (N-1) dimensional space in which is one of the elements. Hence,

................. (6.4.18)

where c_i, i = 1, 2, 3 . . . N-1 are some known constants.

Now consider the equation (6.4.17) to find the error at the n ^th time level

................. (6.4.19)

This error has to be bounded as the number of iterations 'n' increases infinitely, that is as n--->0. From equation (6.4.19) this condition is satisfied whenever maximum of |l_i| < 1 that is the maximum of the absolute value of the eigen values of is less than one. So to check for the stability of any numerical scheme it is sufficient to check for the modulus of the largest eigen value of the coefficient matrix is to be less than unity.

For FTCS scheme the coefficient matrix is

................. (6.4.20)

where I is the identity matrix of order (N-1)*(N-1) and T_N-1 is a square matrix of order (N-1) with eigen values

................. (6.4.21)

and the corresponding eigen vectors are

................. (6.4.22)

From (6.4.20) and (6.4.21) the eigen values of are

................. (6.4.23)

Hence the condition for stability is

Since is always positive, the only useful condition is

................. (6.4.24)

Conditon (6.4.24) is always true whenever that is, the condition for FTCS scheme is to be stable is r must be less or equal to 1/2. This is precisely what has been observed in the example 6.4.1. For this problem if FTCS scheme with r > .5 is used, to obtain the numerical solution the solution became unphysical.

Stability of the Crank-Nicolson scheme:

Consider the Crank-Nicolson scheme to the one-dimensional diffusion equation with zero boundary conditions and some known initial conditions(6.4.7)

for i = 1, 2,.......N-1 and r = 0, 1, 2.................

Therefore the iterative matrix for the present scheme is

and the eigen values of this matrix are

................. (6.4.25)

In(6.4.25) for all positive values of r the denominator is greater than the numerator makes modulus of eigen values of less than one. Hence Crank-Nicolson scheme is unconditionally stable. This is again reflected in the example 6.4.2 because the scheme was giving reasonable results even with r=1.