Peaceman-Rachford (alternating directions) scheme.

n this section we present a general way to construct a finite difference approximation for a solution of the heat equation via a procedure with linear dependency of the amount of computation on the size of the lattice.

We are considering the following boundary problem for the heat equation: MATH MATH MATH MATH

We introduce the uniform lattices MATH MATH the lattice function $u_{ij}^{k}$ and notation . The introduced above notations $\bar{u}$ and $\hat{u}$ refer to the -variable. Consider the scheme

(Alternating directions1)

(Alternating directions2)

in all internal points of the lattice

. The $\Lambda _{x}$ refers to the operator $\Lambda$ acting in the

index. The initial conditions are MATH

and the boundary conditions are

(Alternating boundary1)

(Alternating boundary2)

The key observation about the scheme ( Alternating directions1 )-( Alternating boundary1 ) is that the ( Alternating directions1 ) is a one dimensional implicit scheme in the -direction while the ( Alternating directions2 ) is the implicit scheme in the -direction. Hence, starting from we use ( Alternating directions1 ) to find the $\tilde{u}$ through the factorization procedure of the previous section. Afterwards, we similarly use ( Alternating directions2 ) to find $\hat{u}$ .

To understand the boundary condition ( Alternating boundary1 ) subtract ( Alternating directions1 ) from ( Alternating directions2 ) and obtain

(Alternating boundary)

Notation.

Index.

Contents.