Mathematics / advanced
Numerical ordinary differential equations
Derive reliable integration methods from consistency, stability, stiffness, adaptive error control, and geometric structure.
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Inside this module
Module 01 advanced
Numerical ordinary differential equations
Derive reliable integration methods from consistency, stability, stiffness, adaptive error control, and geometric structure.
- 01
Local and global truncation error
Separate a single exact-start defect from accumulated trajectory error and derive the assumptions behind order claims.
31 min - 02
Runge–Kutta order conditions
Derive low-order conditions by matching solution derivatives and distinguish quadrature accuracy from nonlinear stage coupling.
34 min - 03
Absolute stability regions
Analyze the complex amplification factor of a time step and distinguish boundedness, decay, accuracy, and matrix transient growth.
32 min - 04
Stiff equations and A-stability
Identify stability-limited time stepping and distinguish A-stability, L-stability, damping of fast modes, and temporal resolution.
33 min - 05
Linear multistep methods and root conditions
Analyze consistency and zero-stability of constant-step history formulas before invoking their convergence theorem.
35 min - 06
Embedded pairs and adaptive stepping
Build a scaled local-error controller from shared Runge–Kutta stages and interpret what an embedded difference estimates.
34 min - 07
Implicit solvers and Newton tolerances
Control algebraic error inside an implicit time step using Jacobian conditioning, scaled corrections, and a discretization error budget.
35 min - 08
Dense output and event location
Construct continuous numerical output, locate transverse event roots, and separate interpolation error from root-solver accuracy.
34 min - 09
Symplectic integrators and modified energy
Analyze a structure-preserving oscillator step and distinguish symplectic geometry, exact invariants, and asymptotic modified Hamiltonians.
36 min - 10
Differential-algebraic equations and index
Distinguish implicit ODEs from constrained DAEs, initialize consistent data, and interpret differentiation index under regularity assumptions.
36 min