Numerical methods for partial differential equations

Numerical methods for partial differential equations approximate solutions to equations involving derivatives across space and time, especially when exact solutions are unavailable or impractical.

Connect

Finite difference method: A numerical method that approximates derivatives using differences between function values at discrete points. It replaces derivatives in the PDE with algebraic difference formulas on a grid.
Partial differential equation: An equation relating an unknown function of multiple variables to its partial derivatives. Numerical PDE methods approximate solutions to this broader class of equations.
Computational fluid dynamics: The use of numerical computation to analyze and simulate fluid flow. Fluid-flow models rely on numerical solutions of coupled PDEs such as the Navier–Stokes equations.
Consistency (numerical analysis): The property that a discrete numerical scheme approaches the differential equation as its discretization is refined. It measures whether the chosen discrete equations represent the intended PDE.
Finite element method: A numerical method that approximates solutions with piecewise-defined functions over a mesh and enforces a variational formulation. Its weak-form approach handles complex geometries and boundary conditions.
Initial-boundary value problem: A differential equation problem specified by initial data and conditions on the spatial boundary. Its data define the computational problem that a PDE discretization must represent.
Navier–Stokes equations: Nonlinear PDEs that describe the motion of viscous fluids. Their nonlinear terms and multiple scales make numerical approximation essential in practical flow problems.
Convergence (numerical analysis): The property that numerical solutions approach the exact solution as discretization errors vanish. It is the central test of whether refinement improves a PDE approximation.
Finite volume method: A numerical method that enforces conservation laws over discrete control volumes. Flux balances make it especially effective for conservation-form PDEs.
Boundary condition: A constraint imposed on a solution or its derivatives along the boundary of a domain. Boundary conditions enter the discrete system and can strongly affect accuracy.