Spring 2021, Dr. Vili Heinonen, Dept. of Mathematics.
This is the home page for the 18.303 course at MIT in Spring 2021, 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: TR 9:30-11am https://mit.zoom.us/j/92763506665. Office Hours: Wednesday 9-10am https://mit.zoom.us/j/92763506665.
Grading: 50% homework, 15% mid-term, 35% 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. 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
- Spectral methods for numerically solving PDEs
- Finite difference discretizations
- Properties of Hermitian operators
- (Semilinear) Heat : Equation
- Basic time stepping methods
- Method of Lines (MOL) Solutions
- Lax equivalence, stability, Von Neumann Analysis
- Higher dimensional PDEs
- Generalized boundary conditions
- Separation of Variables
- Wave Equation
- Traveling waves and D'Alembert's solution
- Numerical Dispersion
- Sturm-Liouville Operators
- Distributions
- Green's Functions
- Weak form and Galerkin expansions
- Finite Element Methods
Lecture 1 | Sine series (Julia)
During the first week we covered basic properties of vector spaces and linear operators including norms and inner products. We defined the notion of linear operators and introduced adjoints of linear operators. We went through some examples of smooth functions on the interval [0,1]. We used these tools and notions to solve the Poisson equation with Dirichlet boundary conditions on this interval. We also talked about bases for vector spaces and introduced the notion of an orthogonal and orthonormal bases. We introduced the basic properties of Fourier transform on a finite interval.
Lecture 2 | Finite differences (Julia)
We covered the basic idea of discretizing functions and writing down finite difference approximations of differential operators. We introduced backward, forward, and center difference methods and used these to write a simple discretization for the Laplacian. We talked about matrix representations for the difference operators and the importance of boundary conditions. We briefly discussed how the matrices are invertible if a linear system has a unique solution. As an example we talked about Poisson equation. We also covered deriving finite difference operators using polynomial fitting.
We showed that Laplacian operator is self-adjoint with Dirichlet boundaries. We introduced the notion of positive and negative (semi)definite operators. We talked about the superposition principle and used it to solve the heat equation and the wave equation with Dirichlet boundaries. Important theme during this lecture was the ability to separate partial differential equations in sufficiently symmetric domains.
We discussed some general properties of different boundary conditions for partial differential equations. We showed that the general solution is the solution to the homogeneous problem with the desired boundary condition + the solution to the inhomogeneous problem with zero boundaries. We revisited boundary conditions within the framework of finite difference approximation. We saw how the uniqueness of the solution to a linear PDE given by the boundary condition translates in manifested in the finite difference approximation.
We talked about how some of the ideas we used earlier extend to problems in higher dimensions. The main method here was separation of variables, which is a powerful technique to solve PDEs when the problem is nicely symmetric. This is true especially for time-dependent problems -- time is usually independent of the spatial dimensions so solving for the time evolution once you have solved the spatial part is often not too hard. We also discussed calculating Fourier coefficients in greater detail and defined the finite Fourier transform as a linear map from one vector space to another.
In this lecture we learned some techniques for analyzing the stability of time integration schemes. We considered a wave equation with damping controlled by the coefficient µ. The time evolution of the system can change drastically depending on µ. We briefly talked about what we mean by saying that a time integrator is stable. We saw how we can use the eigenvalues of the Laplacian to analyze the stability of the heat equation with different time discretization schemes.
We talked about general linear PDEs with various boundary conditions. One of the main points was that the solution of an inhomogeneous PDE can be seen as the solution to the homogeneous problem with the given boundary conditions + the solution to the inhomogeneous problem with zero boundaries. We solved the heat equation in n+1 dimensions. If the eigenproblem can be solved for the spatial part, it is always easy to solve the time evolution of the coefficients of the field in the eigenbasis. We showed how to do this.
We also solved the n+1 dimensional wave equation in the eigenbasis of the Laplacian. We introduced the Fourier transform to solve for the time evolution of the coefficients.
We introduced the notion of distributions: a generalization of functions that make sense under integration. Specifically, we discussed Dirac delta distribution that can be used to extract a value of a function at a given point by using an inner product between the Delta distribution and the function. We covered some of the basic features of the delta distribution.
We formally defined the Fourier transform and discussed its properties. We showed how one can see the Fourier transform as a representation of a function in the Fourier basis i.e. as superposition of plane waves. We derived some identities for the Fourier transform using the properties of the delta distribution. We also covered the connection between Fourier coefficients and the Fourier transform.
We introduced Laplace transform and discussed some of its properties (i.e. the connection with the Fourier transform). We talked about the good and the bad of the Laplace transform and pointed out some problems with calculating the inverse Laplace transform. In the end we came up with a reasonable formula for the inverse Laplace transform.
We considered the time evolution of the driven harmonic oscillator (time evolution of the eigenbasis coefficients of the wave equation with an oscillating input). We solved this equation with an ansatz explaining the phenomenon of resonance.
We did a recap of complex analysis: the basic properties of complex numbers, what it means to be complex differentiable and how to do contour integrals on the complex plane. These contour integrals are useful for dealing with (inverse) Fourier and Laplace transforms. We used the Laplace transform to solve the driven harmonic oscillator ODE from last time and covered also the case of resonant driving frequencies.
We discussed the Poisson equation with general Dirichlet boundary conditions. After briefly discussing the 1d case and how we deal with the boundary conditions we formulated the problem in 2d. We showed how to build block matrices that let us write down the Poisson equation as a simple linear system in the discretized setting. We also talked about boundary conditions in the 2d case.
We talked about the weak (variational) formulation of a boundary value problem. The idea is that a certain inner product has to hold for all test functions that go smoothly to zero at the boundary. For computational applications we can use a basis for the function to be solved and the test functions that gives a sparse linear operator for a truncated set of functions. Usually these functions are such that it is easy to calculate their derivatives.
We solved the Laplace equation in spherical coordinates introducing the spherical harmonics and Associated Legendre polynomials. We discussed the properties of these basis functions in the framework of weak solutions to a partial differential equation.
We used the weak (variational) formulation to define Finite Element Methods (FEMs). The basic idea behind FEMs is that we solve the variational problem with an approximate basis of functions that have a small support (set where the functions value is non-zero). Essentially the basis functions are defined on some small elements that can be glued together to form the whole domain of the differential equation. We worked out the Poisson equation in 1d using piecewise linear functionals defined on some intervals and saw that for equally sized elements this gave us essentially the same linear system that the finite difference method gave us. FEMs are especially useful when the domains are complicated and computationally efficient if the solution only varies fast on some small portion of the domain.
We introduced functionals as maps from some vector space to real (or complex) numbers. We wrote them in terms of integrals over functions and their derivatives (these are not all the functionals). We generalized the gradient of a function to a variational (functional) derivative of a functional. This was done using the inner product, which allows us to write the variation of a functional at a given point (given function) to some direction defined by another function. We showed a connection between wave and heat equations to scaler quantities (energies) expressed as functionals.
We discussed Lotka-Volterra system as a simple example of a non-linear system. We talked about stationary points of the system and used perturbative methods to analyze the behavior of the system near these stationary points. In addition we derived a special conserved quantity that can be used to check the solution of the system.
We defined Green's functions as a direct method for solving a linear PDE. A Green's function is essentially the continuous generalization of an inverse of a matrix. We used the delta distribution and Fourier transform to calculate a Green's function explicitly for some model systems. An important takeaway is that Green's functions depend not only on the PDE but also the boundary conditions and the boundary.
We did a quick recap of some of the important results from vector calculus. Most importantly we discussed Gauss' divergence theorem, Stokes' theorem and the fundamental theorem of calculus. All of these theorems relate the integral of some useful differential operator operating on a given function to a certain boundary integral.
We introduced Burgers' equation as a simple advection problem. In order to solve the problem we discussed the method of characteristics: a powerful method for solving 1st order PDEs. We talked about numerical strategies for solving the Burgers' equation and showed that many of the good old finite difference discretization schemes are actually unconditionally unstable. To this end, we introduced upwind and downwind methods. There are implicit or explicit discretization schemes that take into account the direction of the flow.
We also did a quick summary of the different methods covered in this class and discussed strategies for solving different kind of PDEs. We outlined some of the strengths and weaknesses of certain numerical and analytical schemes.