Numerical Partial Differential Equations

Numerical partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).

Numerical techniques for solving PDEs include the following:

  • The finite difference method, in which functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.
  • The method of lines, where all but one variable is discretized. The result is a system of ODEs in the remaining continuous variable.
  • The finite element method, where functions are represented in terms of basis functions and the PDE is solved in its integral (weak) form.
  • The finite volume method, which divides space into regions or volumes and computes the change within each volume by considering the flux (flow rate) across the surfaces of the volume.
  • The spectral method, which represents functions as a sum of particular basis functions, for example using a Fourier series.
  • Meshfree methods don't need a grid to work and so may be better suited for some problems. However the computational effort is usually higher.
  • Domain decomposition methods solve boundary value problems by splitting them into smaller boundary value problems on subdomains and iterating to coordinate the solution between the subdomains.
  • Multigrid methods solve differential equations using a hierarchy of discretizations.

The finite difference method is often regarded as the simplest method to learn and use. The finite element and finite volume methods are widely used in engineering and in computational fluid dynamics, and are well suited to problems in complicated geometries. Spectral methods are generally the most accurate, provided that the solutions are sufficiently smooth.

Numerical partial differential equations
Finite difference methods Heat Equation and related: FTCS scheme · Crank–Nicolson method Hyperbolic: Lax–Friedrichs method · Lax–Wendroff method · MacCormack method · Upwind scheme · Other: Alternating direction implicit method · Finite-difference time-domain method
Finite volume methods Godunov's scheme · High-resolution scheme · MUSCL scheme · AUSM · Riemann solver
Finite element methods hp-FEM · Extended finite element method · Discontinuous Galerkin method · Spectral element method · Meshfree methods · Mortar methods
Other methods Spectral method · Pseudospectral method · Method of lines · Multigrid methods · Collocation method · Level set method · Boundary element method · Immersed boundary method · Analytic element method · Particle-in-cell · Isogeometric analysis
Domain decomposition methods Schur complement method · Fictitious domain method · Schwarz alternating method · Additive Schwarz method · Abstract additive Schwarz method · Neumann–Dirichlet method · Neumann–Neumann methods · Poincaré–Steklov operator · Balancing domain decomposition · BDDC · FETI · FETI-DP

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