Numerical Methods for Partial Differential Equations

1. Numerical Methods for Partial Differential Equations CAAM 452 Spring 2005 Instructor: Tim Warburton

2. Overview • Our final goal is to be able to solve PDE’s of the form: • This is a conservation law with some form of dissipation (under assumptions on A) • We will discuss boundary conditions, solution domain W, and suitable solution spaces for this equation later.

3. Physical Examples • These and similar equations and vector analogs are pervasive: • Fluid mechanics (Euler equations, compressible Navier-Stokes equations, magnetohydrodynamics). • Electromagnetics (Maxwell’s equations) • Heat equation • Shallow water equations • Atmospheric models • Ocean models • Bio-population models (morphogenesis, predator prey, epidemiology) • …..

4. Divide and Conquer • It is highly non-trivial to solve these equations analytically (i.e. with smarts, pen and paper). • We can forget the idea of writing down closed form solutions for the general case. • We will consider the component parts of the equations and discuss techniques to solve the reduced equations. • Some very reduced models admit exact solutions which allow us to check how well we are doing. • Finally we will put different methods together and aim for the big prize.

5. Simplification • Let’s choose a simple example, namely the 1D advection diffusion equation. • This PDE is first order in time and second order in space.

6. Further Simplification • We can simplify even further by dropping the second order diffusion or dissipation term: • This PDE is first order in time and first order in space. • Volunteer to solve this equation analytically?.

7. Necessary Information to Solve The IBVP • The Initial, Boundary, Value Problem represented by the PDErequires some extra information in order to to be solvable. • What do we need?.

8. Answer In this case, because of the hyperbolic nature of the PDE (solution travels from right to left with increasing time), we need to supply: • Extent of solution domain • What is the solution at start of the solution process: u(x,0) • Boundary data: u(b,t) • Final integration time. t As we just sawwe also need tospecify inflow data x=a x=b x Need to specify the solution at t=0

9. Brief Summary • There is a checklist of conditions we will need to consider to obtain a hopefully unique solution of a PDE • The PDE (duh) • Boundary values (also known as boundary conditions) • Initial values (if there is a time-like variable) • Solution domain

10. Periodic Case • Suppose we remove the inflow and imagine that the interval [a,b) is periodic. • Further suppose we wish to solve for the solution at some non-negative time T. • We can indicate this by the following specification:

11. Analytical Solution • Volunteer: • For this PDE to make sense we should discuss something about u0, what?

12. Fourier Series Representation (p4 GKO) In other words, we can express a sufficiently smooth function in terms of an infinite trigonometric polynomial. The fhats are the Fourier coefficients of the polynomial.

13. Returning to the Advection Equation • We wills start with a specific Fourier mode as the initial condition: • We try to find a solution of the same type:

14. cont • Substituting in this type of solution the PDE: • Becomes an ODE: • With initial condition

15. cont • We have Fourier transformed the PDE into an ODE. • We can solve the ODE: • And it follows that the PDE solution is:

16. Note on Fourier Modes • Note that since the function should be 2pi periodic we are able to deduce: • We can also use the superposition principle for the more general case when the initial condition contains multiple Fourier modes:

17. cont • Let’s back up a minute – the crucial part was when we reduced the PDE to an ODE: • The advantage is: we know how to solve ODE’s both analytically and numerically (more about this later on).

18. Add Diffusion Back In • So we have a good handle on the advection equation, let’s reintroduce the diffusion term: • Again, let’s assume 2-pi periodicity and assume the same ansatz: • This time:

19. cont • Again, we can solve this trivial ODE:

20. cont • The solution tells a story: • The original profile travels in the direction of decreasing x (first exponential term) • As the profile travels it decreases in amplitude (second exponential term)

21. What Did Diffusion Do?? • Advection: • Diffusion: • Adding the diffusion term shifted the multiplier on the right hand side of the Fourier transformed PDE (i.e. the ODE) into the left half plane. • We summarize the role of the multiplier…

22. Categorizing a Linear ODE Increasingly oscillatory  Exponential decay Exponential growth  Increasingly  oscillatory Here we plot the behavior of the solution to the top right ODE for mu in the complex plane

23. Solving the Scalar ODE Numerically • We know the solution to the scalar ODE • However, it is also reasonable to ask if we can solve it approximately. • We have now simplified as far as possible. • Once we can solve this model problem numerically, we will apply this technique using the method of lines to approximate the solution of the PDE.

24. ODE Prototype • We will consider ODE’s of the kind

25. ODE Time Stepping Topics We will cover the following details on time stepping the ODE. • Stability of time stepping • Stability regions in the complex plane • Accuracy of time stepping • Convergence of time stepping method with decreasing time step • Examples of explicit time stepping methods: • Euler forward • Leap-frog • Adams-Bashford • Runge-Kutta

26. Reading for Next Week • Study: Gustaffson-Kreiss-Oliger (GKO) p3-17 and p38-39 • Brush up your programming and PDE knowledge – there will be frequent implementation exercises.