Mathematical Physics - Volume II - Numerical Methods
12 Chapter 1. Finite difference method and Finite element method
We can now write the following
u n , j + 1 − u n j k
u t ( x n , t j ) =
+ O ( k ) ,
(1.2)
δ 2
u n + 1 , j − 2 u n j + u n − 1 , j h 2
x u n j h 2
+ O ( h 2 ) ≡
+ O ( h 2 ) ,
u xx ( x n , t j ) =
(1.3)
which introduce the differential difference operator δ 2
x which is analogous to the
differential operator ∂ 2 ∂ x 2 . Equation (1.2) is a two-level equation with respect to t , since it only includes two subsequent values of j . Let area Ω within the plane xt be covered by a mesh ( x n , t j ) . If all derivatives in a partial differential equation , given by:
( x , t ) ∈ Ω .
L [ u ] = f
(1.4)
Replaced with their finite differences, we obtain the finite difference equation: D [ U n j ] = f n j ( x n , t j ) ∈ Ω . (1.5) We can say that equation (1.4) was discretized in order to obtain (1.5). Whose solution, U n j , approximately represents the unknown u ( x , t ) in element nodes ( x n , t j ) 1.1.2 Consistency and convergence In order to obtain good approximation via discretization, the solution of (1.4) needs to be as close as possible to satisfying the condition given by (1.5), for sufficiently small h and k . Local rounding error is represented by the following difference: T n j = D [ u n j ] − f n j . (1.6) Finite difference equation is consistent with the partial differential equation (1.4) under the following condition:
lim k , h → 0
T n j = 0 .
(1.7)
Besides consistency, it is necessary for the approximate solution accuracy to increase when h , k → 0. If U n j is the exact solution of (1.5), and u n j is the solution of (1.4) at point ( x n , t j ) , discretization error is defined as the difference U n j − u n j .. Finite difference method is considered convergent if:
U n j − u n j = 0 ( x n , t j ) ∈ Ω .
lim h , k → 0
(1.8)
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