Numerical Methods for Differential Equations — Homework

Three assignments from the course Numerical Methods for Differential Equations (M.Sc. Mathematical Engineering, University of Padova, 2025/26), covering time integration of ODEs, finite-difference schemes for PDEs and the finite element method. Each folder contains the written report and, where applicable, the MATLAB implementation.

Author: Davide Quartucci

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Contents

#	Topic	Methods	Code
1	ODE time integration	RK4 convergence study; derivation of BDF formulas; stiff linear system from the 2D Laplacian (Crank–Nicolson vs. ode45, error vs. CPU time); Lotka–Volterra with implicit CN + Newton (analytic Jacobian); adaptive-step Crank–Nicolson with error control	MATLAB
2	Finite differences for PDEs	Helmholtz equation with the 5-point Laplacian; Crank–Nicolson for the heat equation; von Neumann stability analysis; schemes for the advection equation	— (analytical)
3	Finite element method	P1 elements for the Laplace/Poisson problem with mixed Dirichlet–Neumann conditions; sparse assembly of the stiffness matrix; L² error over 5 refinement levels; PCG with Jacobi vs. incomplete Cholesky preconditioning	MATLAB

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Highlights

• Convergence verified empirically: log-log error plots compared against the theoretical order of each scheme.
• Stiff problems: explicit vs. implicit integrators compared on accuracy and computational cost.
• Adaptive time stepping: step size controlled by a local error estimate (tolerance 10⁻⁸).
• FEM from scratch: element-by-element assembly into a sparse global matrix, boundary-condition enforcement and preconditioned iterative solvers.

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Repository structure

HW1_RungeKutta/
├── report.pdf
└── matlab_script/        # RK4, ODE systems, Lotka–Volterra, adaptive CN
HW2_FiniteDifferences/
└── report.pdf
HW3_FEM/
├── report.pdf
└── matlab_script/
    ├── main.m                          # driver: loops over mesh levels 0–4
    ├── assemble_system.m               # stiffness matrix assembly
    ├── apply_boundary_conditions.m     # Dirichlet / Neumann enforcement
    └── Inputfiles/                     # meshes (coord, topol, bound, trace) + reference solution

How to run

Open MATLAB in the relevant matlab_script/ folder and run the scripts. For HW3, run main.m and unzip Inputfiles/solRef.dat.zip first.

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Relevance to quantitative finance

The same toolkit underlies PDE-based derivative pricing: the Black–Scholes equation is a parabolic PDE solved with Crank–Nicolson finite-difference schemes, and stability, convergence and the efficiency of linear solvers determine the accuracy and speed of a pricer.