Finite difference

Finite volume

• Godunov
• High-resolution
• Monotonic upstream-centered (MUSCL)
• Advection upstream-splitting (AUSM)
• Riemann solver
• essentially non-oscillatory (ENO)
• weighted essentially non-oscillatory (WENO)

Finite element

• hp-FEM
• Extended (XFEM)
• Discontinuous Galerkin (DG)
• Spectral element (SEM)
• Mortar
• Gradient discretisation (GDM)
• Loubignac iteration
• Smoothed (S-FEM)

Meshless/Meshfree

• Smoothed-particle hydrodynamics (SPH)
• Moving particle semi-implicit method (MPS)
• Material point method (MPM)

Domain decomposition

• Schur complement
• Fictitious domain
• Schwarz alternating
• • additive
  • abstract additive
• Neumann–Dirichlet
• Neumann–Neumann
• Poincaré–Steklov operator
• Balancing (BDD)
• Balancing by constraints (BDDC)
• Tearing and interconnect (FETI)
• FETI-DP

Others

• Spectral
• Pseudospectral (DVR)
• Method of lines
• Multigrid
• Collocation
• Level-set
• Boundary element
• Immersed boundary
• Analytic element
• Particle-in-cell
• Isogeometric analysis
• Infinite difference method
• Infinite element method
• Galerkin method