PARTIAL DIFFERENTIAL EQUATIONS ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier Dr. B.A. DeVantier

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Partial Differential Equations An equation involving partial derivatives of an unknown function of two or more independent variables The following are examples. Note: u depends on both x and y

Partial Differential Equations An equation involving partial derivatives of an unknown function of two or more independent variables The following are examples. Note: u depends on both x and y

Partial Differential Equations Because of their widespread application in engineering, our study of PDE will focus on linear, second-order equations The following general form will be evaluated for B2 - 4AC

Partial Differential Equations Because of their widespread application in engineering, our study of PDE will focus on linear, second-order equations The following general form will be evaluated for B2 - 4AC

B2-4AC Category Example < 0 Elliptic Laplace equation (steady state with 2 spatial dimensions = 0 Parabolic Heat conduction equation (time variable with one spatial dimension >0 Hyperbolic Wave equation (time-variable with one spatial dimension

y or t set up a grid estimate the dependent variable at the center x

y or t set up a grid estimate the dependent variable at the center or intersections of the grid x

Scope of Lectures on PDE Finite Difference: Elliptic The Laplace Equation Finite difference solution Boundary conditions Finite Difference: Parabolic Heat conduction Explicit method Simple implicit method

Specific Study Objectives Recognize the difference between elliptic, parabolic, and hyperbolic PDE Recognize that the Liebmann method is equivalent to the Gauss-Seidel approach for solving simultaneous linear algebraic equations Recognize the distinction between Dirichlet and derivative boundary conditions

Specific Study Objectives Know the difference between convergence and stability of parabolic PDE Understand the difference between explicit and implicit schemes for solving parabolic PDE

Finite Difference: Elliptic Equations B2- 4AC < 0 Typically used to characterize steady-state boundary value problems Before solving, the Laplace equation will be solved from a physical problem

Elliptic Equations B2- 4AC < 0 Typically used to characterize steady-state boundary value problems Before solving, the Laplace equation will be solved from a physical problem

The Laplace Equation Models a variety of problems involving the potential of an unknown variable We will consider cases involving thermodynamics, fluid flow, and flow through porous media

The Laplace equation Let’s consider the case of a plate heated from the boundaries How is this equation derived from basic concepts of continuity? How does it relate to flow fields?

Consider the plate below, with thickness Dz. The temperatures are known at the boundaries. What is the temperature throughout the plate? T = 400 T = 200 T= 200 T = 200

First, recognize how the shape can be set in an x-y coordinate system

Divide into a grid, with increments by Dx and Dy

What is the temperature here, if using a block centered scheme? y T = 400 T = 200 T= 200 T = 200 x

What is the temperature here, if using a grid centered scheme? y T = 400 T = 200 T= 200 T = 200 x

Consider the element shown below on the face of a plate D z in thickness. The plate is illustrated everywhere by at its edges or boundaries, where the temperature can be set. y D y x D x

q(y + D y) q(x) q(x + D x) Consider the steady state heat flux q in and out of the elemental volume. q(y) By continuity, the flow of heat in must equal the flow of heat out.

Rearranging terms

Again, this is our continuity equation Dividing by Dx, Dy, Dz and D t : As x & y approach zero, the equation reduces to: Again, this is our continuity equation

Equation A The link between flux and temperature is provided by Fourier’s Law of heat conduction Equation B where qi is the heat flux in the direction i. Substitute B into A to get the Laplace equation

Equation A Equation B

Consider Fluid Flow In fluid flow, where the fluid is a liquid or a gas, the continuity equation is: The link here can by either of the following sets of equations: The potential function: Stream function:

The Laplace equation is then

Flow in Porous Media Darcy’s Law The link between flux and the pressure head is provided by Darcy’s Law

For a case with sources and sinks within the 2-D domain, as represented by f(x,y), we have the Poisson equation. Now let’s consider solution techniques.

Evaluate these equations based on the grid and central difference equations (i,j+1) (i,j) (i-1,j) (i+1,j) (i,j-1)

we can collect the terms to get: (i,j) (i+1,j) (i-1,j) (i,j+1) (i,j-1) If D x = D y we can collect the terms to get:

This equation is referred to as the Laplacian difference equation. (i,j) (i+1,j) (i-1,j) (i,j+1) (i,j-1) This equation is referred to as the Laplacian difference equation. It can be applied to all interior points. We must now consider what to do with the boundary nodes.

Boundary Conditions Dirichlet boundary conditions: u is specified at the boundary Temperature Head Neumann boundary condition: the derivative is specified qi Combination of both u and its derivative (mixed boundary condition)

The simplest case is where the boundaries are specified as fixed values. This case is known as the Dirichlet boundary conditions. u2 u1 u3 u4

Consider how we can deal with the lower node shown, u1,1 Note: This grid would result in nine simultaneous equations. u1 1,2 u3 1,1 2,1 u4 -4u1,1 +u1,2+u2,1+u1 +u4 = 0

Let’s consider how to model the Neumann boundary condition centered finite divided difference approximation (0,100) (0,0) (200,0) (200,100) h = 0.05x + 100 y x suppose we wanted to consider this end grid point

The two boundaries are consider to be symmetry lines due to the fact that the BC translates in the finite difference form to: h i+1,j = h i-1,j and h i,j+1 = h i,j-1 1,2 1,1 2,1

h1,1 = (2h1,2 + 2 h2,1)/4 h1,2 = (h1,1 + h1,3+2h22)/4 2,2 1,2 1,1 2,1

EXAMPLE The grid on the next slide is designed to solve the LaPlace equation Write the finite difference equations for the nodes (1,1), (1,2), and (2,1). Note that the lower boundary is a Dirichlet boundary condition, the left boundary is a Neumann boundary condition, and Dx = Dy.

The Liebmann Method Most numerical solutions of the Laplace equation involves systems that are much larger that the general system we just evaluated Note that there are a maximum of five unknown terms per line This results in a significant number of terms with zero’s

The Liebmann Method In addition to the fact that they are prone to round-off errors, using elimination methods on such sparse system waste a great amount of computer memory storing zeros Therefore, we commonly employ approaches such as Gauss-Seidel, which when applied to PDEs is also referred to as Liebmann’s method.

The Liebmann Method In addition the equations will lead to a matrix that is diagonally dominant. Therefore the procedure will converge to a stable solution. Over relaxation is often employed to accelerate the rate of convergence

As with the conventional Gauss Seidel method, the iterations are repeated until each point falls below a pre-specified tolerance:

Groundwater Flow Example

Modeling 1/2 of the system shown, we can develop the following schematic where D x = D y = 20 m (0,100) (0,0) (200,0) (200,100) h = 0.05x + 100 y x The finite difference equations can be solved using a a spreadsheet. This next example is part of the PDE example you can download from my homepage.

Secondary Variables Because its distribution is described by the Laplace equation, temperature is considered to be the primary variable in the heated plate problem A secondary variable may also be of interest In this case, the second variable is the rate of heat flux across the place surface

FINITE DIFFERENCE APPROXIMATION BASED ON RESULTS OF TEMPERATURE DISTRIBUTION

THE RESULTING FLUX IS A VECTOR WITH MAGNITUDE AND DIRECTION Note: q is in degrees If qx=0, q is 90 or 270  depending on whether qy is positive or negative, respectively

Finite Difference: Parabolic Equations B2- 4AC = 0 These equations are used to characterize transient problems. We will first study this in one spatial direction then we will discuss the results in 2-D.

Consider the heat-conduction equation As with the elliptic PDEs, parabolic equations can be solved by substituting finite difference equations for the partial derivatives. However we must now consider changes in time as well as space

y x t x { temporal { spatial

Forward finite divided difference Centered finite divided difference

We can further reduce the equation: NOTE: Now the temperature at a node is estimated as a function of the temperature at the node, and surrounding nodes, but at a previous time

EXAMPLE Consider a thin insulated rod 10 cm long with k = 0.835 cm2/s Let D x = 2 cm and D t = 0.1 sec. At t=0 the temperature of the rod is zero. hot cold

Convergence and Stability Convergence means that as D x and D t approach zero, the results of the numerical technique approach the true solution Stability means that the errors at any stage of the computation are attenuated, not amplified, as the computation progresses The explicit method is stable and convergent if

Derivative Boundary Conditions To TL In our previous example To and TL were constant values. However, we may also have derivative boundary conditions Thus we introduce an imaginary point at i = -1 This point provides the vehicle for providing the derivative BC

Derivative Boundary Conditions q0 = 0 TL For the case of qo = 0, so T-1 = T1 . In this case the balance at node 0 is:

Derivative Boundary Conditions q0 = 10 TL For the case of qo = 10, we need to know k’ [= k/(C)]. Assuming k’ =1, then 10 = - (1) dT/dx, or dT/dx = -10

Derivative Boundary Conditions

Implicit Method Explicit methods have problems relating to stability Implicit methods overcome this but at the expense of introducing a more complicated algorithm In this algorithm, we develop simultaneous equations

Explicit

Explicit Implicit

Explicit Implicit grid point involved with space difference

Explicit Implicit grid point involved with space difference grid point involved with time difference

Explicit Implicit grid point involved with space difference grid point involved with time difference With the implicit method, we develop a set of simultaneous equations at step in time

which can be expressed as: For the case where the temperature level is given at the end by a function f0 i.e. x = 0

Substituting In the previous example problem, we get a 4 x 4 matrix to solve for the four interior nodes for each time step