Haar wavelet approach to linear stiff systems

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Abstract

A simple and effective algorithm based on Haar wavelet is proposed to the solution of linear stiff problems in this paper. And it can integrate the stiff equation with very accurate results for any length of time. The simulation result shows that the single-term Haar wavelet method (STHW) is better than the classical Runge–Kutta fourth-order method (CRK), while the terms of the both expansions are the same.

Introduction

When the solution of a system contains components which change at significantly different rates for given changes in the independence variable the system is said to be “stiff” [5], [12]. In practical solution of stiff problems, the choice of the solution steps is critical. Large steps will lose some fast changing properties of the system, while small steps will introduce too many round-off errors and cause numerical instability.

In [2], Carroll presents an exponentially fitted scheme for solving the stiff systems of initial-value problems. A long-standing work by Pavlov and Rodionova [13], [14] retains nonlinear terms of the original equations. However, these methods cannot reduce computational effort. In this paper, we solve the linear stiff systems via Haar wavelets by taking advantage of some nice properties of Haar wavelets.

Section snippets

Preliminary works

The orthogonal set of Haar wavelets hi(t) is a group of square waves with magnitude of ±1 in some intervals and zeros elsewhere [3], [4], [6], [7], [8], [9], [10], [11]. In general, hn(t)=h1(2jt−k),n=2j+k,j≥0,0≤k<2j,n,j,k∈Z,h1(t)=1,0≤t<12−1,12≤t<1.Namely, each Haar wavelet contains one and just one square wave, and is zero slsewhere. Just these zeros make Haar wavelets to be local and very useful in solving stiff systems.

Any function y(t), which is square integrable in the interval [0,1], can

Solution of linear stiff system

Consider a linear stiff system of the following form: ẋ(t)=Ax(t)+Bu(t),x(0)=x0,t∈[0,1/m],In the normalized interval with τ=m·t ẋ(τ)=1m[Ax(τ)+Bu(τ)],x(0)=x0,τ∈[0,1],Now expressing ẋ(τ), x(τ), and u(τ) in single-term Haar series as ẋi(τ)=aih0(τ),xi(τ)=bih0(τ),ui(τ)=rih0(τ),and the following recursive relationship is obtained with P=1/2, bi=12ai+xi(j−1),xi(j)=ai+xi(j−1)=2bi−xi(j−1),where i,j=0,1,⋯,n∈N. Substituting , into Eq. (13), we obtain nonlinear algebraic equations and they may be

Numerical examples

Consider linear stiff systems in the following examples.

Example 1

This example is due to Alt [1]ẋ1(t)ẋ2(t)=−195−1−97x1(t)x2(t),x(0)=11.

The exact solution is x1(t)=147(95exp(−2t)−48exp(−96t))x2(t)=147(48exp(−96t)−exp(−2t)).

Example 2

We take the stiff system of Rosenbrock and Storey [17]ẋ1(t)ẋ2(t)=−100000.909−1x1(t)x2(t),x(0)=10.999.

The exact solution is x1(t)=exp(−1000t)x2(t)=−0.909999exp(−1000t)+998.91999exp(−t).

Example 3

Consider the following linear system [12]ẋ1(t)ẋ2(t)ẋ3(t)=−0.1−49.900−500070−120x1(t)x2(t)x3(t),x

Additional comparisons

We would like to point out that this is a common property for many transform approaches to linear systems including Haar, Walsh, block pulse, Fourier and many others; and all result in a generalized Lyapunov equation. The kernel of the complete solution lies on how to solve this generalized Lyapunov equation. Compare the proposed STHW approach with the known approaches as follows.

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To be submitted to Mathematics and Computers in Simulation. This research was supported in part by the National Science Council of Taiwan under Grant NSC.

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