numerical solutions of burgers' equation project report
TRANSCRIPT
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)
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
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.
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.
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.
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.
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.
2. MATLAB CODE AND EXAMPLES
Differential Equation:
1. y’’=x+1
y(0)=0, y(1)=1
Solution:
Y0 0
Y1 14/81
Y2 40/81
Y3 1
2. y’’(x) + y(x) + 2x(1-x) = 0
y(0) = y(1) = 0
Y0 0
Y1 0.05017
Y2 0.05102
Y3 1
Galerkin Solution
Analytic Solution
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
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
THANK YOU !