Mathematics / advanced
Numerical partial differential equations
Compare discretizations of distributed systems using consistency, stability, conservation, adaptivity, and solver costs.
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Inside this module
Module 01 advanced
Numerical partial differential equations
Compare discretizations of distributed systems using consistency, stability, conservation, adaptivity, and solver costs.
- 01
The method of lines and semidiscrete systems
Separate spatial approximation from time integration and measure stability in the appropriate discrete norm.
30 min - 02
Finite-difference stencils and numerical dispersion
Read a stencil as a Fourier multiplier and separate phase error from amplitude error.
30 min - 03
Von Neumann stability and CFL conditions
Analyze amplification on periodic constant-coefficient grids and keep geometric CFL necessity separate from sufficient stability.
30 min - 04
Finite-volume conservation and numerical fluxes
Build updates from shared interface fluxes and separate conservation from entropy and positivity.
30 min - 05
Riemann solvers and high-resolution reconstruction
Combine entropy-selected interface states with slope-limited reconstruction while tracking spatial and temporal accuracy separately.
30 min - 06
Spectral methods and aliasing
Track DFT normalization, resolvable modes and nonlinear frequency folding before claiming high-order accuracy.
30 min - 07
Discontinuous Galerkin fluxes and stability
Join elementwise weak equations through oriented traces and derive a sign-consistent semidiscrete energy balance.
30 min - 08
Operator splitting and splitting error
Approximate a combined evolution by ordered subflows and measure the noncommutativity that remains.
30 min - 09
Multigrid and smoothing properties
Combine high-frequency damping with coarse correction rather than mistaking either operation for a complete solver.
30 min - 10
A posteriori error estimation and adaptive meshes
Estimate discretization error with element residuals and flux jumps while preserving the assumptions behind reliability.
30 min