(Prereq checkpoint: comfortable with multivariable calculus, linear algebra, ODEs, complex numbers/Fourier series. Light exposure to distributions & basic functional analysis will help from Stage 3 onward.)
Goal. Build intuition via canonical PDEs and boundary-value problems; master eigenfunction expansions.
Primary texts (focus chapters).
- Haberman, Applied PDEs with Fourier Series and BVPs: Ch. 1–5 (Fourier series; heat; wave; Laplace; Sturm–Liouville).
- Strauss, PDE: An Introduction: Ch. 2–6 (classification; heat/wave/Laplace; Sturm–Liouville; basics of Green’s functions).
- (Quick on-ramp: Farlow, PDE for Scientists and Engineers: early chapters on separation & Fourier.)
Must-know.
- PDE classification (elliptic/parabolic/hyperbolic); well-posedness intuition.
- Separation of variables; superposition; orthogonality & completeness.
- Fourier series; Sturm–Liouville problems; eigenfunction expansions.
- Canonical solutions: heat (diffusion), wave (d’Alembert in 1D), Laplace/Poisson in rectangles/disks (at least conceptually).
Nice-to-know.
- Bessel/Legendre families; cylindrical/spherical domains.
- Handling inhomogeneous BCs/forcing; non-rectangular geometries via coordinate transforms.
- Maximum principles for Laplace/heat (statement and use).
Project.
- Vibrating membrane modes. Compute & visualize eigenmodes of a rectangular (and circular, if possible) membrane. Implement separation of variables → transcendental eigenvalue conditions → truncation & reconstruction; animate time evolution from arbitrary initial data.
Goal. Solve IVPs on unbounded/half-space domains; internalize convolution and imaging.
Primary texts (focus chapters).
- Haberman: Ch. 6–8 (Fourier transform; Laplace transform; delta functions/DFT/FFT).
- Strauss: Green’s functions & method of images (around Ch. 7); first-order transport/characteristics (Ch. 1).
- McOwen, PDE: Methods and Applications: transform methods & Green’s functions (corresponding chapters).
Must-know.
- Fourier transform solution of heat/wave/Poisson on (\mathbb{R}^n); convolution kernels (heat kernel, Poisson kernel).
- Laplace transform for linear IVPs; transfer-function view.
- Method of images; half-space Green’s functions; basic reflection arguments.
- First-order linear PDEs via method of characteristics.
Nice-to-know.
- Distributional Fourier transform; Plancherel/Parseval (form and use).
- Hankel transform for radial problems; Poisson summation (as intuition).
- Asymptotics/steepest descent (idea level) for wave propagation.
Project.
- Green’s function Poisson solver. Construct a 2D free-space Poisson solver via FFT-based convolution; extend to half-space with method of images. Validate against finite-difference solutions on a large box.
Goal. Move from classical to weak solutions; derive FEM from energy/variational principles.
Primary texts (focus chapters).
- Evans, PDE: Ch. 5 (Sobolev spaces), selected parts of Ch. 6–7 (weak formulation; Lax–Milgram; elliptic theory).
- Renardy & Rogers, An Introduction to PDEs: distributions, Fourier transform, and weak formulations (early/middle chapters).
- Larson & Bengzon, The Finite Element Method (open): intro to variational forms & (H^1) FEM.
- Logg–Mardal–Wells, Automated Solution of PDEs by FEM (FEniCS Book): practical weak forms → code.
Must-know.
- Weak derivatives; Sobolev spaces (H^1, H_0^1, H^{-1}); Poincaré inequality.
- Variational formulation of Poisson & linear elasticity; energy methods.
- Lax–Milgram theorem; existence/uniqueness for coercive elliptic problems.
- Galerkin idea; Céa’s lemma; interpolation/error norms.
Nice-to-know.
- Trace theorem; compact embeddings; regularity (elliptic (H^{2}) on smooth domains).
- Fredholm alternative; mixed formulations (e.g., Poisson in (H(\text{div}))).
- Isoparametric elements; curved boundaries; conditioning & preconditioning basics.
Project.
- FEM Poisson in 2D. Derive weak form; implement linear-triangle FEM on a polygonal domain with Dirichlet BCs. Verify order of convergence on a manufactured solution; add Neumann BC; solve a simple elasticity problem (plane stress) if time permits. (Fast path: prototype in FEniCS; deeper path: C++ with Eigen + your own mesh/assembly.)
Goal. Become fluent with discretization choices and stability/accuracy trade-offs.
Primary texts (focus chapters).
- Strikwerda, Finite Difference Schemes and PDEs: consistency–stability–convergence; von Neumann analysis; heat/wave/ADI.
- LeVeque, Finite Volume Methods for Hyperbolic Problems: Ch. 1–6 (conservation laws; Riemann problems; Godunov; high-resolution schemes).
- Trefethen, Spectral Methods in MATLAB: core chapters on Chebyshev differentiation, spectral advection/heat/Poisson.
- (Alternative overview: Morton & Mayers, Numerical Solution of PDEs.)
Must-know.
- Truncation error; consistency; Lax equivalence; von Neumann analysis.
- Diffusion: FTCS/BTCS/Crank–Nicolson; stability & damping; method of lines.
- Wave/transport: upwind vs central; dispersion/dissipation; CFL.
- Conservation laws: Godunov/Rusanov; limiters (minmod, MC); MUSCL idea.
- Spectral collocation (Chebyshev/Fourier); aliasing & dealiasing (2/3-rule); Poisson via spectral methods.
Nice-to-know.
- ADI schemes; operator splitting; IMEX.
- Nonuniform meshes; ghost-cell BCs; immersed boundaries (idea level).
- Multigrid V-cycle for Poisson; domain decomposition; spectral element overview.
Project.
- Compare three discretizations. Solve 1D Burgers’ equation (viscous & inviscid limits) using: (a) 2nd-order upwind FV with limiter, (b) Crank–Nicolson FD (viscous), (c) Fourier spectral method. Measure accuracy vs cost; visualize shock formation and Gibbs/limiter behavior.
Goal. Tackle physically relevant nonlinear models; connect analysis, numerics, and modeling.
Primary texts (focus chapters).
- LeVeque, Finite Volume Methods for Hyperbolic Problems: nonlinear Riemann solvers; entropy; high-res methods (later chapters).
- Smoller, Shock Waves and Reaction–Diffusion Equations: traveling waves & stability (selected sections).
- Murray, Mathematical Biology I: reaction–diffusion pattern formation (selected chapters).
- Salsa, PDE in Action: applied models across physics/engineering; readable case studies.
- (Fluids/CFD bridge: Chorin–Marsden or Taira’s notes on projection methods, as references.)
Must-know.
- Nonlinear characteristics; rarefaction/shock; weak solutions & entropy conditions.
- Reaction–diffusion systems: diffusion-driven instability (Turing), traveling waves, basic spectral stability.
- Incompressible Navier–Stokes at moderate Re: nondimensionalization, projection method (Chorin), pressure Poisson equation, boundary conditions.
- Parameter & model calibration from data; verification/validation mindset.
Nice-to-know.
- Hamilton–Jacobi and viscosity solutions (conceptual).
- High-order WENO/ENO; TVD Runge–Kutta; positivity-preserving tweaks.
- Phase-field/Cahn–Hilliard; porous-medium equation; basic optimal control with PDE constraints.
Project.
- 2D Gray–Scott simulator (reaction–diffusion) with finite differences or FEM. Explore parameter space to reproduce known patterns; compute dispersion relation around steady states to link numerics with linear stability. Stretch goal: Implement a 2D lid-driven cavity at low–moderate Re with a projection method; validate against benchmark streamfunction/vorticity values.
- One core book per stage (bolded above), with a second as cross-reference for problem sets.
- Implement as you read. For every analytic technique, code a minimal solver and a regression test on a manufactured solution.
- Quantify error. Always plot error vs grid/time step; confirm observed orders; track stability limits.
- Archive results. Keep a small “PDE gallery” repo: short notebooks/programs, problem statements, and validation plots for each project.
If you want, I can turn this into a printable checklist or map the projects to C++/Eigen + Python/FEniCS starter templates.
https://chatgpt.com/share/691244f9-ea34-800d-b674-a8bf844ee2ca