On the derivation of the nonlinear discrete equations numerically integrating the Euler PDEs

F N Koumboulis,M G Skarpetis,B G Mertzios

doi:10.1155/s102602269800003x

Abstract

The Euler equations, namely a set of nonlinear partial differential equations (PDEs), mathematically describing the dynamics of inviscid fluids are numerically integrated by directly modeling the original continuous-domain physical system by means of a discrete multidimensional passive (MD-passive) dynamic system, using principles of MD nonlinear digital filtering. The resulting integration algorithm is highly robust, thus attenuating the numerical noise during the execution of the steps of the discrete algorithm. The nonlinear discrete equations approximating the inviscid fluid dynamic phenomena are explicitly determined. Furthermore, the WDF circuit realization of the Euler equations is determined. Finally, two alternative MD WDF set of nonlinear equations, integrating the Euler equations are analytically determined.

Highlights

The problem of deriving the analytic solution of nonlinear (NL) partial differential equations (PDEs) is a rather difficult task
One important set of NLPDEs is that of the Euler equations
The Euler equations describe the dynamics of an inviscid fluid, having a variety of applications in fluid mechanics

Summary

INTRODUCTION

The problem of deriving the analytic solution of nonlinear (NL) partial differential equations (PDEs) is a rather difficult task. The discretization of ODEs (Ordinary Differential Equations) via WDFs appears to be a wellestablished technique since the beginning of 70s (see [1 1] and the references therein) The advantage of this technique is mainly based on the preservation of passivity when the continuous system is transformed to the respective discrete approximation. This way the numerical errors in the iteration of the steps of the discrete algorithm, simulating the original system, appear to be attenuated [12] This distinct advantage of the WDFs has been transferred to the case of deriving approximate discretizations of PDEs [13,14,15]. The contribution of the present paper can be summarized into the analytic determination of the nonlinear discrete set of equations approximating the Euler equations for a polytropic fluid and the respective MD WDF equations integrating the Euler equations. For presentation purposes the three equivalent expressions will be partitioned into three different subsections

Generalized Time

Normalization

Hadamard Transformation

DISCRETIZATION ALGORITHM

WDF REALIZATION

CONCLUSIONS