Discretizations

Discretization is the process of approximating a continuous mathematical model by a discrete one that can be solved with a computer. It replaces continuous variables, domains, and operators with a finite set of quantities, a mesh or grid, and approximate operators. The aim is to produce a solvable system whose solution converges to the original problem as the discretization becomes finer.

Discretizations introduce discretization error and require a balance between accuracy, stability, and computational cost. Key concepts include the choice of grid or mesh, the representation of unknowns, and the treatment of boundary and initial conditions. Time discretization converts evolution problems into sequences of algebraic problems with a chosen time step.

Finite difference methods approximate derivatives by difference quotients on a grid. They are simple and effective for regular domains but can be less flexible for complex geometries.

Finite element methods formulate problems variationally and approximate the solution by piecewise basis functions over a mesh. FEM excels in complex geometries and heterogeneous materials.

Finite volume methods preserve integral quantities and fluxes across control volumes, making them well suited for conservation laws and fluid dynamics. Spectral methods use global basis functions, achieving high accuracy for smooth problems but with limitations on geometry and discontinuities.

Discretization quality is assessed via consistency, stability, and convergence. Refining the mesh or decreasing time steps improves accuracy but increases cost. Discretization is central to numerical simulation in engineering, physics, and applied mathematics.