Fall 2019, Dr. Chris Rackauckas, Dept. of Mathematics.

This is the home page for the 18.303 course at MIT in Spring 2019, where the syllabus, lecture materials, problem sets, and other miscellanea are posted.

Course description

Provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Studies operator adjoints and eigenproblems, series solutions, Green's functions, and separation of variables. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems, including stability and convergence analysis and implicit/explicit timestepping. Julia (a Matlab-like environment) is introduced and used in homework for simple examples.

Prerequisite: linear algebra (18.06, 18.700, or equivalent).

Lectures: MWF 1–2pm (2-146). Office Hours: Wednesday 2–3pm (2-347).

Grading: 45% homework, 25% mid-term (March 20th), 30% final project (due the last day of class). Problem sets are due in class on the due date. The lowest problem set score will be dropped at the end of the term. Missed midterms require a letter from Student Support Services or Student Disabilities Services to justify accommodations. Legitimate excuses include sports, professional obligations, or illness. In the event of a justified absence, an alternative make-up project will be assigned.

Collaboration policy: Make an effort to solve the problem on your own before discussing with any classmates. When collaborating, write up the solution on your own and acknowledge your collaborators.

Books: Introduction to Partial Differential Equations by Olver.

Final project: There is a final project instead of a final exam. In your project, you should consider a PDE or possibly a numerical method not treated in class, and write a 5–10 page academic-style paper that includes:

Review: why is this PDE/method important, what is its history, and what are the important publications and references? (A comprehensive bibliography is expected: not just the sources you happened to consult, but a complete set of sources you would recommend that a reader consult to learn a fuller picture.) Analysis: what are the important general analytical properties? e.g. conservation laws, algebraic structure, nature of solutions (oscillatory, decaying, etcetera). Analytical solution of a simple problem. Numerics: what numerical method do you use, and what are its convergence properties (and stability, for timestepping)? Implement the method (e.g. in Julia) and demonstrate results for some test problems. Validate your solution (show that it converges in some known case).

You must submit a one-page proposal of your intended final-project topic, summarizing what you intend to do, by Wednesday, March. 13th. Some suggestions of possible projects will be given before then.

• Why PDEs are interesting.
• The Fourier series and eigenfunction expansions for the Poisson equation
• Optional: Julia Tutorial (5-7PM 32-141)
• Spectral methods for numerically solving PDEs (Problem Set 1)
• Finite difference discretizations
• Properties of Hermitian operators (Problem Set 2)
• (Semilinear) Heat : Equation
• Basic time stepping methods
• Method of Lines (MOL) Solutions (Problem Set 3)
• Lax equivalence, stability, Von Neumann Analysis
• Higher dimensional PDEs
• Generalized boundary conditions (Problem Set 4)
• Advection Equation
• Upwinding operators, Lax-Wendorff
• (Brief!) 1D conservation laws and Berger's Equation (Problem Set 5)
• Separation of Variables
• Wave Equation
• Traveling waves and D'Alembert's solution
• Numerical Dispersion
• Sturm-Liouville Operators (Problem Set 6)
• Distributions
• Min-Max Theorem
• Green's Functions (Problem Set 7)
• Stochastic (Partial) Differential Equations and Diffusion-Advection Equations
• Weak form and Galerkin expansions
• Finite Element Methods

In lecture one we reviewed linear algebra to build a perspective for infinite dimensional linear algebra. Topics like "What is a vector space?" were revisited and the abstraction away from real numbers to an algebraic structure was emphasized. Questions as to the difference between a matrix and a linear operator were settled. With this in mind, we introduced linear algebra on functions. The space of smooth functions forms a vector space. What are some linear operators for smooth functions? What is a basis for smooth functions? How do you write the infinite matrix for the derivative operator in the polynomial basis? We then used these ideas to solve the Poisson equation u_xx = f by diagonalizing the derivative operator and getting a solution in terms of sine functions.

Lecture Notes || Problem Set 1

In this lecture we revisit Fourier series and make the ideas more concrete. We prove some properties of the Fourier series and transform and then write down the logic for how we solve the Poisson equation using operator notation. Using this formalism, we continue onto the Heat equation u_t = u_xx + f and show how to solve the Heat equation by using the eigenfunction basis of the Laplacian. Some simple facts about the Heat equation are then revealed by this solution and plotted.

Lecture Notes

In this lecture some computational mathematics tools were demonstrated. Lyx, Jupyter notebooks, and Weave.jl were shown as options for writing mathematical documents and incorporating numerical results. A recap of the Heat equation was shown, and an emphasis on problem conversion was given. The same tools were then used to solve the Wave Equation. Then moved from global (spectral) bases to forming a discrete local basis. For this basis, we took evenly-spaced points at which to represent the function. The forward and central difference approximations were derived and order of convergence was discussed.

Lecture Notes

In this lecture we continued our discussion of differencing operators, proving the second order convergence of the second derivative central differencing operator. Alternative derivations from polynomial interpolation and Fornberg's algorithm were discussed. From there, we looked at the properties of the discretization matrices and identified similarities with the Fourier case, leading to the idea of self-adjoint operators as a generalization of symmetric to infinite bases.

Lecture Notes

In this lecture we continued the discussion of self-adjoint operators and found it to be a bit more nuanced than we thought. The issue, was boundary conditions. Thus, we worked to clarify the standard boundary conditions (Dirchlet, Neumann, Robin), and showed how this effects the properties of the "linear" operator. With some boundary conditions the operator was no longer linear!

Lecture Notes Problem Set 2

In this lecture we began to explore the convergence of numerical methods on the Heat Equation. First we derived the forward time centered space and backwards time centered space approximations by using our previous work on finite difference operators. We showed that we could alternatively think about the discretization of these PDEs as a two part problem: discretizing just space gives an ODE that can then be analyzed. From here, the Lax Equivalence Theorem was introduced, and the consistency of these methods was demonstrated. But are they stable?

In this lecture we went over the stability of methods for ODEs. For an ODE u'=f(u), the method locally behaves like u'=Ju where J is the Jacobian, and its divergence essentially depends on u'=lambda*u where lambda is the maximal eigenvalue. Thus we make this our test equation for stability and see what is the maximal dt for a method at which the test equation goes to zero. When plotting this stability region we could see the stepsize limits of various methods and compare them.

In this lecture we went over the stability of methods for PDEs. One way is to use the ODE method, but that requires knowing the eigenvalues of the matrices. A more direct way to analyze the stability was introduced: Von Neumann analysis. Here we used a Fourier mode decomposition of the numerical solution and calculated the growth factor w.r.t. each of the modes. This gave a condition on when the different methods would be stable. It turns out that the implicit methods (shown here) were more stable than the explicit methods, giving them a reason to be used even though they are more costly for the same amount of error.

Lecture Notes

In this lecture we showed how to generalize the finite differencing methods to multiple dimensions using the Kron operator, and how to compactly solve 2D equations using the non-commutivity of matrices.

Lecture Notes

In this lecture we went over the advection equation and derived the first derivative stencils. We showed that the methods must be moving in the same direction as the true solution in order to be stable, giving the upwinding methods. The higher order upwinding methods were shown to be derived from polynomial approximations.

Lecture Notes Problem Set 3

Here we went over numerical linear algebra from the perspective of numerical PDEs. Wilson's algorithm shows that we can make O(n) methods for solving implicit systems with tridiagonal matrices, an important fact since our stencils give tridiagonal matrices! However, to handle more general cases, we dove into matrix factorizations, iterative methods, and eventually (preconditioned) Krylov subspace methods (from a high level, proofs withheld). Together these give methods which greatly enhance the speed of implicit methods

Lecture Notes

In this lecture we went over the common methods for solving timestepping a method of lines (MOL) discretization of PDE. We started with the BDF method and described in detail how the adaptive BDF2 method was hotstarted, how the error is approximated, and how the time step is adapted. Additionally, it was discussed how the nonlinear iterations were performed without requiring matrix inversions. Afterwards, newer methods for solving PDEs were discussed. IMEX (E)SDIRK and exponential integrators were described, and the advantages were shown. Pseudospectral discretizations were introduced. At the end, improvements to the matrix factorization routines were shown, starting with the ADI method which split Crank-Nicholson into two parts and in doing so greatly simplified the linear algebra. This was shown to be a more general partial factorization scheme that could be used to speed up other methods like BDF2.

• notes on the min–max theorem
• min–max Julia examples

In order to get an intuitive feel for what the eigenfunctions of self-adjoint operators should look like, a powerful tool is the min–max theorem. See notes and examples.

Further reading: See, for example the min-max theorem and the Rayleigh quotient articles in Wikipedia, although this presentation is rather formal. Unfortunately, much of the discussion you will find of this principle online and in textbooks is either (a) full of formal functional analysis or (b) specific to quantum mechanics [where the operator is Â=-∇²+V for some "potential-energy" function V(x)].

Extra Credit Problem Set

This extra credit problem set has two computing exercises that go through pseudospectral methods and writing an implicit time stepping function. Since these are important tools in the practical research of PDEs in applications, students should be familiar with these concepts.