Numerical analysis Time-dependent partial differential equations Concept
Finite difference method
Summary
The finite difference method solves differential equations approximately on a grid of points. It replaces each derivative by a difference quotient of the values at nearby grid points, which turns the equation into a system of algebraic equations. It is the oldest and simplest numerical method for partial differential equations. Its convergence rests on consistency and stability, as in the Lax equivalence theorem. For time-dependent problems, stability often requires a step restriction such as the Courant–Friedrichs–Lewy condition.
Where this leads
16 pages build on this one.
- von Neumann stability analysis
- Minimum degree algorithm
- Lax–Friedrichs method
- Godunov's theorem
- MacCormack method
- Method of lines
- Alternating-direction implicit method
- Nested dissection
- Multigrid method
- Lax equivalence theorem
- Immersed boundary method
- Courant–Friedrichs–Lewy condition
- Upwind scheme
- Lax–Wendroff method
- Crank–Nicolson method
- Numerical diffusion
Further reading
About this page
| Tag | X7HYEBA5SA |
|---|---|
| URI | m2://X7HYEBA5SA |
| Kind | concept |
| Writing | W0 stub |
| MSC | 65M06, 65N06 |
| Same as |
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Definitions
Results about this
- von Neumann stability analysis
- Godunov's theorem
- Lax equivalence theorem
- Courant–Friedrichs–Lewy condition
Used by
- von Neumann stability analysis
- Lax–Friedrichs method
- Godunov's theorem
- MacCormack method
- Method of lines
- Alternating-direction implicit method
- Nested dissection
- Multigrid method
- Lax equivalence theorem
- Immersed boundary method
- Courant–Friedrichs–Lewy condition
- Upwind scheme
- Lax–Wendroff method
- Crank–Nicolson method
- Numerical diffusion