Project 4
Burger's Equation
Group Members: Patrick Bishop, Robert Argus, and Vincent Caetto
The general form for the code we used was: Burger's Equation
Nonlinear Partial Differential Equations are PDEs whose boundary conditions contain functions whose derivatives depend
on more than one independent variable. These types of equations are found in many different fields of science, such as
fluid mechanics, thermodynamics and relativity, among others. As NPDEs contain partial derivatives that rely on more than
one variable, we cannot simply solve them using the "centered-difference" formulas. Instead, we have to apply finite
difference approximations to the individual partial derivates. By using the Newton's Method, we can calculate the
"backward-difference" equations and evaluate them at each step to find the corresponding values for our variables at said
step. An example of a nonlinear partial differential equation is Burger's equation.
Reconstruction of Example 8.12: Solving Burger's Equation
Burger's Equation is an elliptic equation describing a simplified model for fluid flow. It is given by:
\[u_t + uu_x = Du_{xx}\]
It is evident that Burger's Equation is nonlinear, as the \(uu_x\) term relies on a product of the function and a
partial derivative. We can use Burger's Equation to form a set of boundary equations for our nonlinear PDE:
\[\begin{cases} u_t + uu_x = Du_{xx} \\ u(x,0) = f(x) \\ u_x(x_l,t) = l(t) \\ u_x(x_r,t) = r(t) \end{cases}\]
This set of conditions is known as Dirichlet boundary conditions. Using these conditions, we can then use Newton's method
to get:
\[ \frac{w_{ij}-w_{i,j-1}}{k} + w_{ij}\frac{w_{i+1,j}-w_{i-1,j}}{2h} - D\frac{w_{i+1,j}-2w_{ij}+w_{i-1,j}}{h^2} = 0\]
This "backward-difference" equation lets us solve for \(w_{ij}\),\(w_{i+1,j}\) and \(w_{i-1,j}\) at each time step j.
The j-1 refers to the data from the previous step, letting us extend the time to any limit we choose. We can also rewrite
this equation, by setting \(\vec{Z} = [w_{ij},...,w_{mj}]\), where m is the number of unknowns. Using this conversion,
we get:
\[ \vec{Z}^{k+1} = \vec{Z}^k - DF(\vec{Z}^k)^{-1}F(\vec{Z}^k) \]
By iterating this equation, we can find our final values for each \(w_{ij}\).
Exercise 8.4.3: Solutions to Burger's Equation
Computer Problem 8.4.1: Equilibrium Solutions to Burger's Equation
Computer Problem 8.4.2: Burger's Equation with Homogeneous Dirichlet Boundary Conditions
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