Quantitative Analysis.
Trading Platform.
Python for Excel.
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I.Basic math.
II.Pricing and Hedging.
III.Explicit techniques.
IV.Data Analysis.
V.Implementation tools.
1.Finite differences.
A.Finite difference basics.
B.One dimensional heat equation.
C.Two dimensional heat equation.
a.The Peaceman-Rackford (alternating directions) scheme.
b.Stability of Peaceman-Rackford.
D.General techniques for reduction of dimensionality.
E.Time dependent case.
2.Gauss-Hermite Integration.
3.Asymptotic expansions.
4.Generation of random samples.
5.Monte-Carlo.
6.Convex Analysis.
VI.Basic Math II.
VII.Implementation tools II.
Bibliography.
Forum Notation Index Contents

The Peaceman-Rackford (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:MATHMATHMATHMATH

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

MATH(Alternating directions1)
MATH(Alternating directions2)
 in all internal points of the lattice MATH. The $\Lambda _{x}$ refers to the operator $\Lambda$ acting in the $i$ index. The initial conditions areMATH and the boundary conditions are
MATH(Alternating boundary1)
MATH(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 $x$-direction while the ( Alternating directions2) is the implicit scheme in the $y$-direction. Hence, starting from $u$ 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

MATH(Alternating boundary)




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