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

2. CAAM 452 Spring 2005 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 , and suitable solution spaces for this equation later.

3. CAAM 452 Spring 2005 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. CAAM 452 Spring 2005 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. CAAM 452 Spring 2005 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. CAAM 452 Spring 2005 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. CAAM 452 Spring 2005 Necessary Information to Solve The IBVP The Initial, Boundary, Value Problem represented by the PDE requires some extra information in order to to be solvable. What do we need?.

8. CAAM 452 Spring 2005 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: a)Extent of solution domain b)What is the solution at start of the solution process: u(x,0) c)Boundary data: u(b,t) d)Final integration time. x t Need to specify the solution at t=0 As we just saw we also need to specify inflow data x=ax=b

9. CAAM 452 Spring 2005 Brief Summary There is a checklist of conditions we will need to consider to obtain a hopefully unique solution of a PDE 1)The PDE (duh) 2)Boundary values (also known as boundary conditions) 3)Initial values (if there is a time-like variable) 4)Solution domain

10. CAAM 452 Spring 2005 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. CAAM 452 Spring 2005 Analytical Solution Volunteer: For this PDE to make sense we should discuss something about u 0, what?

12. CAAM 452 Spring 2005 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. CAAM 452 Spring 2005 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. CAAM 452 Spring 2005 cont Substituting in this type of solution the PDE: Becomes an ODE: With initial condition

15. CAAM 452 Spring 2005 cont We have Fourier transformed the PDE into an ODE. We can solve the ODE: And it follows that the PDE solution is:

16. CAAM 452 Spring 2005 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. CAAM 452 Spring 2005 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. CAAM 452 Spring 2005 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. CAAM 452 Spring 2005 cont Again, we can solve this trivial ODE:

20. CAAM 452 Spring 2005 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. CAAM 452 Spring 2005 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. CAAM 452 Spring 2005 Categorizing a Linear ODE Exponential growth  Exponential decay Increasingly oscillatory  Increasingly  oscillatory Here we plot the behavior of the solution to the top right ODE for mu in the complex plane

23. CAAM 452 Spring 2005 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. CAAM 452 Spring 2005 ODE Prototype We will consider ODE’s of the kind

25. CAAM 452 Spring 2005 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. CAAM 452 Spring 2005 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.