Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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. Peaceman-Rachford (alternating directions) scheme.
b. Stability of Peaceman-Rachford.
D. General techniques for reduction of dimensionality.
E. Time dependent case.
2. Gauss-Hermite Integration.
3. Asymptotic expansions.
4. Monte-Carlo.
5. Convex Analysis.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

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 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 are MATH 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)

Notation. Index. Contents.

Copyright 2007