What Are Numerical Models?

Numerical models are mathematical representations of physical, chemical, biological, or engineering systems that are solved computationally using discrete approximations rather than closed-form analytical solutions. They translate the governing equations of a phenomenon, typically differential or integral equations describing conservation of mass, energy, momentum, or charge, into systems of algebraic equations that a computer can solve over a discretized domain. The approach is necessary because the vast majority of real-world systems involve geometries, boundary conditions, or nonlinearities that preclude exact analytical treatment.

The discipline draws on applied mathematics, computer science, and the domain science being modeled. Key inputs are the governing physics (expressed as partial differential equations), the computational grid, material or system parameters, and boundary and initial conditions. Key outputs are field quantities such as temperature distribution, stress, fluid velocity, or electromagnetic potential, resolved to the scale of the discretization grid.

Finite Difference and Finite Element Methods

Three families of discretization underlie most numerical models in engineering. The finite difference method (FDM) replaces derivatives in a PDE with difference quotients evaluated on a structured grid, making it the most direct path from equation to computation and well suited to regular geometries such as atmospheric grids in weather modeling. The finite element method (FEM) subdivides the domain into elements of arbitrary shape, approximates the solution within each element using polynomial basis functions, and assembles a global sparse matrix system. As the COMSOL reference on the finite element method describes, FEM handles complex geometries and multi-physics coupling that resist structured-grid approaches, which is why it dominates structural mechanics, heat transfer, and electromagnetics. The finite volume method (FVM) enforces conservation laws at the cell level by integrating flux across each cell face, making it the standard approach in computational fluid dynamics, where mass and momentum conservation are paramount. As Machine Design's comparison of FEM, FDM, and FVM summarizes, method selection depends on geometry complexity, the governing physics, and the desired balance between implementation effort and accuracy.

Discretization and Grid Generation

The accuracy and computational cost of a numerical model are both controlled by the resolution of the underlying grid or mesh. Structured grids, which organize nodes in regular rows and columns, are computationally efficient and support high-order difference schemes, but they cannot conform to irregular boundaries without coordinate transformations. Unstructured meshes composed of triangles, tetrahedra, or mixed element types adapt to curved geometries but require more sophisticated solvers. Adaptive mesh refinement refines the grid automatically in regions where the solution gradient is large, concentrating computational effort where accuracy matters most and coarsening it elsewhere. Grid convergence, the property that the solution changes negligibly as the grid is refined, is a prerequisite for trusting a numerical result.

Validation and Verification

Verification and validation (V&V) are the formal processes that assess whether a numerical model is solving the right equations correctly. Verification asks whether the code solves its mathematical formulation without programming errors, typically by comparing to manufactured solutions or problems with known analytical answers. Validation asks whether the mathematical model accurately represents the physical system it claims to simulate, typically by comparing model predictions against experimental measurements. As research on verification, validation, and sensitivity studies in computational biomechanics published through NIH/PMC illustrates, the two processes address different error sources: code errors versus model-form errors. Uncertainty in numerical models can be categorized as numerical uncertainty, which decreases with grid refinement, and model-form uncertainty, which reflects missing physics and does not improve with finer grids.

Applications

Numerical models have applications in a range of fields, including:

• Structural and mechanical engineering for stress, fatigue, and vibration analysis
• Computational fluid dynamics for aircraft aerodynamics, turbomachinery, and chemical reactors
• Weather forecasting and climate simulation using atmospheric and oceanic models
• Electromagnetic modeling for antenna design, radar cross-section prediction, and circuit simulation
• Biomedical engineering for blood flow, tissue mechanics, and radiation dose planning