What Is Scientific Computing?

Scientific computing is a discipline concerned with the development and application of computational methods to solve quantitative problems in science and engineering. It occupies the intersection of numerical mathematics, computer science, and application domain expertise, translating physical theories and mathematical models into algorithms that can be executed on digital hardware to produce predictions, design parameters, or data analyses that are otherwise inaccessible through experiment or closed-form mathematics alone. Problems addressed by scientific computing range from the simulation of airflow over an aircraft wing to the folding of protein chains to the propagation of seismic waves through the Earth's crust.

The field draws on numerical analysis for the mathematical theory of algorithms, on parallel computing architectures for the hardware platforms that execute large-scale simulations, and on domain sciences for the physical equations that the algorithms must approximate. Its outputs are numerical solutions to differential equations, optimization problems, and data decomposition tasks that carry quantifiable bounds on their accuracy.

Numerical Methods and Algorithms

The algorithmic core of scientific computing consists of methods for solving the classes of mathematical problems most common in physical science: systems of linear and nonlinear equations, ordinary and partial differential equations, eigenvalue problems, and numerical integration. Finite difference, finite element, and spectral methods discretize continuous differential equations into systems of algebraic equations that computers can solve directly. Iterative linear solvers such as the conjugate gradient method and the generalized minimal residual method handle the sparse matrix systems that arise in large-scale discretization. The design of numerical algorithms for modern hardware is detailed in research by Dongarra, Grigori, and Higham on numerical algorithms for high-performance computational science, which addresses how mixed-precision arithmetic, communication-avoiding formulations, and randomized methods extend algorithm efficiency on contemporary supercomputer architectures.

High-Performance Computing Infrastructure

Scientific computing at scale requires parallel computer systems whose many processors work concurrently on different portions of a calculation. Distributed-memory clusters, in which processors communicate by passing messages using the Message Passing Interface standard, handle problems too large for a single node's memory. Shared-memory multicore systems and graphics processing units accelerate the dense linear algebra and stencil computation kernels common in partial differential equation solvers. The Society for Industrial and Applied Mathematics publishes instructional texts on high-performance scientific computing that cover parallel programming models including OpenMP, MPI, and GPU programming frameworks such as CUDA and OpenCL. Exascale systems, capable of 10^18 floating-point operations per second, present new algorithmic challenges in fault tolerance, energy efficiency, and data movement that the field is actively addressing.

Simulation and Modeling

Scientific simulation is the use of validated computational models to replicate the behavior of physical systems under conditions that cannot be studied directly through experiment, either because of scale, cost, hazard, or physical inaccessibility. Climate models solve coupled atmospheric and oceanic fluid dynamics equations over global grids and are among the most computationally intensive scientific applications. Molecular dynamics simulations track the positions and velocities of millions of atoms over nanosecond timescales to study material properties and biological processes. Computational fluid dynamics models resolve turbulent flows around engineered structures. A review of numerical analysis and algorithms for high-performance computing compiled in MDPI Mathematics outlines the range of contemporary simulation methods and identifies open research problems in algorithm design for heterogeneous computing platforms.

Applications

Scientific computing has applications in a wide range of disciplines, including:

• Aerospace engineering, through computational fluid dynamics simulations for aerodynamic design
• Climate science and weather forecasting, through numerical integration of atmospheric models
• Drug discovery, through molecular dynamics and quantum chemistry calculations of protein-ligand binding
• Structural engineering, through finite element analysis of mechanical stress in materials and structures
• Nuclear energy, through neutron transport simulations for reactor design and safety analysis