Mathematical Physics - Volume II - Numerical Methods

26 Chapter 1. Finite difference method and Finite element method

solution U exists, then the following holds:

T C

= U T ( A ′ C ) = 0 ,

B ′ C = ( A ′ U )

(1.73)

which suggests that the sum of B ′ terms is zero. This condition is equivalent to: Z Ω f ( x , y ) d x d y = 0 ,

(1.74)

if the integral is calculated using the trapeze rule in nine introduced nodal points (fig. 1.1) Thus, this condition ( R Ω f d Ω = R S g d s = 0) is sufficient for the existence of a (non-unique) solution of the system A ′ U = B ′ . 1.1.10 Curvilinear boundaries We will now demonstrate how to apply finite differences to the following problem: u xx + u yy = f ( x , y ) nad Ω (1.75) u = g ( x , y ) na S (1.76) in the case that S represents a curvilinear boundary of domain Ω . In each node of mesh defined in Ω , which is surrounded by four nodes also within the same domain, usual expressions for finite difference method are applied. Let us now consider nodes for which at least one adjacent node is not in Ω , e.g. P = ( x m , y n ) , fig. 1.2.

y

r

q

P

Q

R

h

h

x

Figure 1.2: Finite difference mesh for curvilinear domain.

Coordinates o points r and q on contour S , obtained by drawing a horizontal and an orthogonal line through P , are ( x m + α h , y n ) and ( x m , y n + β h ) , respectively, where 0 < α , β < 1. Since u is known at S , u ( r ) and u ( q ) are also known. By deriving into a Taylor series, we obtain:

α h 2 2

3 ) ,

u ( q ) = u ( P )+ α hu x ( P )+

u xx ( P )+ O ( h

(1.77)

h 2 2

3 ) .

u ( Q ) = u ( P ) − hu x ( P )+

u xx ( P )+ O ( h

(1.78)

By summing we obtain:

α u ( Q ) − ( 1 + α ) u ( P )+ u ( q ) h 2 α ( α + 1 ) / 2

u xx ( P ) =

+ O ( h ) .

(1.79)