Mathematical Physics - Volume II - Numerical Methods

1.1 Finite difference method

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1.1.3 Stability

Let U n j be the solution of equation (1.5), with initial values of U n 0 and certain boundary values. Let V n j be the solution of a system of finite difference equations, which differs from (1.5) only in the sense of initial values, i.e. the relation V n 0 ≡ U n 0 + E n 0 , holds, where E n 0 is the "error", i.e. the initial difference (deviation). It can be show that E n 0 propagates with an increase in j , towards a homogeneous partial differential equation, with given homogeneous boundary conditions:

D [ E n j ] = 0 .

When the finite difference equation (1.5) is applied to approximate determining of u ( x , T ) for a fixed T = t o + jk , it is clear that h , k → 0 requires that j → ∞ . In addition, in order for equation (1.5) to be applicable to a fixed mesh for the purpose of approximate determining of u ( x n , t j ) , for increasing t j , it is once again necessary for j → ∞ . For partial equations with limited solutions, it is said that the solution of (1.5) is stable if E n j is uniformly limited along n when j → ∞ , i.e. when the following holds: | E n j | < M ( j > J ) (1.9) where M is an arbitrary constant, and J is a positive integer. If h and k are functionally dependent in order to ensure that (1.9) is fulfilled, than the solution of the finite difference equation is conditionally stable. The stability of this solution also implies its convergence. Let us also define the matrix criteria for stability. For this purpose, we will con sider a boundary problem with the initial condition including N nodes along the x di rection, and let us define a vector-column of errors at level j , E j = ( E 1 j , . . . , E N j ) T . For a two-level finite difference method, errors at levels j and j + 1 are related by the following expression: E j + 1 = CE j , (1.10) where C is a matrix of N × N order. Let ρ ( C ) , the spectral radius of C , denote the highest eigenvalue of matrix C . In this case, the matrix criteria of stability can be defined as: a two-level finite difference method for a boundary problem with an initial condition with a limited solution is stable (in matrix terms) if ρ ( C ) ≤ 1. Matrix criteria is a necessary condition for two-level finite difference method stability, but also becomes a sufficient condition if C is symmetrical or nearly symmetrical, whereas all of its eigenvalues are real.