A Class of Block Multi-derivative Numerical Techniques for Singular Advection Equations

M. O. Ogunniran

doi:10.46481/jnsps.2019.12

Abstract

The integration of some differential equations is hard to acquire because of the presence of singular point(s) in these equations. These equations are best solved by some unique technique. Multi-derivative techniques have a long history of powerful integration of such equations yet till date, a couple of class of this technique has been explored for integrating partial differential equations. This work centers around the development, analysis, and implementation of a class of multi-derivative technique on partial differential equations. The approaches were effectively analyzed and were turned out to be consistent, stable and convergent. Numerical outcomes got likewise demonstrated the approximation quality of the technique over existing techniques in the literature.

Highlights

IntroductionConsidering an equation which is known as the linear advection equation for a quantity u(x, t): ∂u ∂t + v(x, t) ∂u ∂x = (1)
Considering an equation which is known as the linear advection equation for a quantity u(x, t): ∂u ∂t + v(x, t) ∂u ∂x = (1)where v(x, t) is known, This type of equation in (1) is classified as first order hyperbolic partial differential equation
Equation (1) is possibly the simplest partial differential equation, this simplicity is deceptive in the sense that it can be very difficult to integrate numerically due to the presence of singularity, a distinctive feature of first order hyperbolic partial differential equations

Summary

Introduction

Considering an equation which is known as the linear advection equation for a quantity u(x, t): ∂u ∂t + v(x, t) ∂u ∂x = (1). Where v(x, t) is known, This type of equation in (1) is classified as first order hyperbolic partial differential equation. Equation (1) is termed as a conservation law since it expresses conservation of mass, energy or momentum under the conditions for which it is derived, i.e. the assumptions on which the equation is based. Equation (1) is possibly the simplest partial differential equation, this simplicity is deceptive in the sense that it can be very difficult to integrate numerically due to the presence of singularity, a distinctive feature of first order hyperbolic partial differential equations. Equation (1) will have to be supplemented with an initial condition: u(x, 0) = u0(x) (2)

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Discussion

Conclusion