Published on July 14, 2016
1. END SEMESTER REPORT ON Numerical Solutions of Burgers’ Equation By Shikhar Agarwal (2012B4A8613P) and Keshav Raheja (2012B4A8678P) M.Sc. (Hons) Mathematics and B.E. (Hons) Electronics and Instrumentation Engineering Submitted in complete fulfillment of the MATHS F266 (Study Project) To Dr. Devendra Kumar BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, PILANI (April, 2015)
2. TABLE OF CONTENTS Acknowledgement Abstract 1. Introduction 1.1 Finite Element Method (An overview ) 1.2 Choosing the basis functions 1.3 Galerkin method approximations 2. MATLAB Codes and Examples 3. Method of Quasi-Linearization for solving PDEs 4. Conclusions References
3. ACKNOWLEDGEMENT We are grateful to our institute, BITS Pilani for providing us the opportunity to pursue a Study Oriented Project in Mathematics. I would like to express a deep sense of gratitude and thanks to Dr. Dilip K. Maiti without whom it would have been impossible to complete the report in this manner. We are also indebted to Dr. Devendra Kumar for his invaluable guidance and support throughout the project without which the successful execution of this project would not have been possible.
4. ABSTRACT This report provides a broad introduction to one of the most popular and widely used numerical methods to solve second order linear boundary value problems: the finite element method, which is in turn used to solve various non-linear PDEs one of which is Burgers’ Equation The report draws from numerous case studies and applications which were implemented by using MATLAB as a computational tool in order to calculate the error between the exact and approximate solution by Galerkin Finite Element Method.
5. 1. INTRODUCTION Numerical analysis is the area of mathematics and computer science that creates, analyses, and implements algorithms for obtaining numerical solutions to problems involving continuous variables. Such problems arise throughout the natural sciences, social sciences, engineering, medicine, and business. The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to problems which usually cannot be solved by analytical methods, the variety of which is suggested by the following: Advanced numerical methods are essential in making numerical weather prediction feasible. Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations. Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving partial differential equations numerically Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. Historically, such algorithms were developed within the overlapping field of operations research In our report, we will be taking up one of the most important numerical method – Finite Element Method. 1.1 FINITE ELEMENT METHOD ( An Overview ) Consider a one-dimensional problem where f is given, u is an unknown function of x , and u’’ is the second derivative of u with respect to x. The first and foremost step is to devise the weak formulation of a given problem.
6. The weak form of P1 If u solves P1, then for any smooth function v that satisfies the displacement boundary conditions, i.e. v = 0 at x = 0 and x = 1, we have --------------- (1) Conversely, if with u(1) = u(0) = 0 satisfies (1) for every smooth function v(x) then one may show that this will solve P1. By using integration by parts on the right-hand-side of (1), we obtain ----------- (2) where we have used the assumption that v(0) = v(1) = 0. 1.2 CHOOSING THE BASIS FUNCTIONS To complete the discretization, we must select a basis of . In the one- dimensional case, for each control point xK we will choose the piecewise linear function vk in whose value is 1 at xk and zero at every xj for every j not equal to k i.e., for k = 1,2,…..,n where each of these functions is known as Hat function, popularly known triangular hat function.
7. 1.3 GALERKIN METHOD APPROXIMATIONS Now, we substitute the obtained system of basis functions in (2) and get a system of equations which when solved simultaneously renders a Tridiagonal stiffness matrix. The error in the solution exponentially decreases as the number of nodal points in increased. Also, contrary to the Finite Difference Method, in FEM, we can find the value of the unknown function at any point in the constructed interval, in addition to the nodal points.
8. 2. MATLAB CODE AND EXAMPLES
9. Differential Equation: 1. y’’=x+1 y(0)=0, y(1)=1 Solution: Y0 0 Y1 14/81 Y2 40/81 Y3 1
10. 2. y’’(x) + y(x) + 2x(1-x) = 0 y(0) = y(1) = 0 Y0 0 Y1 0.05017 Y2 0.05102 Y3 1
11. Galerkin Solution Analytic Solution
12. 3. METHOD OF QUASILINEARIZATION TO SOLVE BURGERS’ EQUATION The inviscid Burgers' equation is a conservation equation, more generally a first order quasilinearhyperbolic equation. In fact by defining its current density as the kinetic energy density: it can be put into the current density homogeneous form: . The solution of conservation equations can be constructed by the method of quasi-linearization. This method yields that if X(t) is a solution of the ordinary differential equation. then U(t) = u[ X(t) , t ] is constant as a function of . For Burgers equation in particular [ X(t), U(t) ] is a solution of the system of ordinary equations: The solutions of this system are given in terms of the initial values by
13. 4. CONCLUSION For boundary value problems having no exact solution we approximate the solution by numerical methods. In this report the finite element method was considered and via various case studies its accuracy in predicting a numerical solution close to the exact solution was established. In our MATLAB code, we have used a very famous algorithm for computing matrices inverses, namely, Thomas Algorithm. For this purpose MATLAB was used as a computational tool. References Differential Equation with boundary value problems by Dennis Zill A modern introduction to differential equations by H.Richardo Differential Equations with applications and historical notes by G.F.Simmons
14. THANK YOU !
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