A computational technique based on asymptotic analysis for solving singularly perturbed ODE systems involving a small parameter

Singularly perturbed problems involving a small parameter are ubiquitous in the field of engineering and present a formidable challenge in their numerical solution. For example, in the field of fluid mechanics such problems appear in the form of convection-diffusion partial differential equations and are used, in the computation of temperature in compressible flows, to describe the model equations for concentration of pollutants in fluids and to describe the momentum relation of the Navier-Stokes equations. In the field of electrical engineering an important application is the telegraph equation describing the voltage or the current as a function of time and position along a cable. The discretization of such systems usually leads to stiff systems of ordinary differential equations (ODEs) containing a small parameter. Such equations are usually stiff and require special care in their numerical solution. On the other hand, such systems can be treated by methods based on asymptotic analysis which renders a reasonable approximate decomposition of the original system into two new systems, for the slow mode and fast modes, respectively, which are no longer stiff. Of special interest are asymptotic methods based on the Chapman-Enskog procedure (CEP) in which the bulk part of the slow mode is left unexpanded. This technique is especially popular amongst mathematical physicists interested in obtaining the diffusion approximation for a wide range of evolution equations, for example, the Boltzmann equation, telegraph equation, and Fokker-Plank equation [

Consider the initial value problem

The method proposed in this paper aims to avoid the difficulties presented by the preceding two approaches. Here, we follow the algorithm of the asymptotic expansion first proposed by Mika and Palczewski [

In order to keep the present exposition self-contained, we consider some aspects of the derivation of the asymptotic method presented in [

Using the initial conditions

The asymptotic convergence of the approximate solutions derived above is proved in [

In the standard first-order approach described, for example, in [

Consider the boundary value problem:

The boundary value problem (

The initial-value problem has a uniquely determined solution

In this paper instead of applying the shooting method to the stiff second-order system of equations (

Here we choose

According to the new approach we solve (

The graphical solutions depicted in Figure

Figure

Red (dashed):

Comparison of Error as a function of

Since this paper is concerned with systems of ODEs, the following example is directed to such a case. The technique is illustrated for a system of coupled ODEs by choosing

Errors in

time | ||
---|---|---|

One has

Choose

Numerical solution

Numerical solution

Numerical solution

We consider the following boundary layer problem from fluid mechanics:

Choose

Numerical solution

Consider the singularly perturbed telegraph equation:

As in the previous example we used the method of lines to obtain the discretized system of ODEs:

Errors for various solution components: red (dotted) using (

The numerical solution of singularly perturbed second-order ODEs of the form (

The authors would like to thank Professor Janusz Mika for providing them with useful suggestions during the preparation of this manuscript. Also, the authors would like to extend their appreciation to the anonymous referees, whose feedback contributed in improving the manuscript.