Volume : VI, Issue : IX, September - 2017

A STUDY ON NUMERICAL ANALYSIS USING BOUNDARY VALUE PROBLEMS

Lavenya K, M. Karthigeyan

Abstract :

 The original motivation behind this thesis was to construct embedded Runge–Kutta methods for use in computing numerical solutions to hyperbolic conservation laws. The methods would use a high–order linearly stable Runge–Kutta scheme in smooth regions of the spatial domain and, in the vicinity of shocks or other discontinuities, switch to a lower–order scheme possessing a nonlinear stability" property which would help prevent spurious oscillations and overshoots. The derivation of such a method turned out to be more challenging and interesting than was originally thought and, as such, this thesis has more to do with the construction of these embedded methods than it does with the original motivational example.

This chapter begins with an introduction to Runge–Kutta methods and linear stability. It then touches iey on the topics related to the solution of hyperbolic conservation laws, including nonlinear stability and strong–stability–preserving Runge–Kutta schemes. Finally, the chapter concludes with a discussion of linearly stable Runge–Kutta methods with embedded strong–stability–preserving Runge–Kutta schemes.

One of the earliest mathematical writings is a Babylonian tablet form the Yale Babylonian collection (YBC 7289), which gives a hexadecimal approximation of the length of the diagonal in a unit square. Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in astronomy, carpentry and construction.

Then, in the Euler’s method is reviewed. The derivation of Euler’s method is stated and using some basic definitions, error. After completing this, we will view the Runge Kutta Method of order four (RK4), we present some necessary definitions of the stability, and examined the absolute stability of Fourth Order Runge–Kutta method. Then, we give the general form of system for first order equations. The aim of this study is to compute the approximated solutions of system of differential equations. Then, it follows the stability theory for systems. Finally, some examples are given, and each numerical method is solved manually. After that, the results of numerical methods are analyzed. Finally, we concluded by giving the advantages and disadvantages. And the Runge kutta method is wide–used in solving ordinary differential equations, and it is more accurate than the Eul            er method. Then, the error between Euler Explicit Method and Runge Kutta Method compared with the Exact Method.

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Article: Download PDF   DOI : 10.36106/ijsr  

Cite This Article:

Lavenya K, M.Karthigeyan, A STUDY ON NUMERICAL ANALYSIS USING BOUNDARY VALUE PROBLEMS, INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCH : Volume-6 | Issue-9 | September-2017

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