Numerical and Computational Mathematics

Numerical and computational mathematics develops and analyses algorithmic strategies for approximating solutions to mathematical models that lack closed-form expressions. Central themes include the discretisation of continuous operators—translating partial and ordinary differential equations into algebraic systems via finite-difference, finite-element or spectral methods—and the design of efficient solvers for the resulting linear or nonlinear systems. Time-integration schemes, from classical Runge–Kutta to multistep and operator-splitting methods, address stability and accuracy in stiff or multi-scale problems. Integral equations are tackled using collocation, Galerkin or Nyström approaches, often within boundary-element contexts that demand careful treatment of singular kernels. Modern advances exploit isogeometric analysis and mesh adaptivity to represent complex geometries exactly and concentrate computational effort where error indicators peak. In parallel, powerful iterative linear-algebra techniques—Krylov subspace methods, multigrid, preconditioning—enhance scalability on parallel architectures. Across all methodologies, rigorous error estimates and convergence theory underpin reliability, while applications range from fluid dynamics and materials modelling to optimisation, data assimilation and uncertainty quantification in engineering and the physical sciences.

Research from Nature Portfolio

A novel coupling of finite-element and finite-difference discretisations has been devised for the Richards equation governing unsaturated–saturated subsurface flow. By employing isogeometric finite elements in horizontal directions and classical finite differences vertically, the method delivers high accuracy and strong parallel scalability in regional hydrological simulations.

A time-consistent stabilisation strategy for incompressible flows replaces the conventional timestep-dependent Petrov–Galerkin parameter with a physical time scale derived from local acceleration and velocity norms. The resulting finite-element formulation remains mathematically consistent as timesteps diminish and substantially reduces pressure errors in cardiovascular benchmark problems, including fluid–structure interaction with compliant walls.

High-resolution shock-capturing techniques have been extended to three-phase immiscible flow in porous media. An explicit finite-volume discretisation of mass-conservation laws incorporates nonlinear capillary and permeability effects, accurately resolving sharp saturation fronts under discontinuous capillary-pressure relations and facilitating long-duration contaminant transport studies on multicore platforms.

Research from all publishers

The virtual element method has been advanced through projection-based discretisations on general polygonal and polyhedral meshes. By avoiding explicit shape functions and employing tailored Gram-Schmidt projections, the approach achieves optimal polynomial convergence orders while accommodating arbitrary element geometries without loss of accuracy.

Norm-resolvent convergence theory for higher-order elliptic differential–difference operators with small variable translations has been established under broad boundary conditions. Precise estimates for resolvent operators and spectral stability as translation parameters vanish provide rigorous guarantees for nonlocal perturbations in advanced material and wave-propagation models.

A comprehensive survey of the spectral finite-element method highlights its deployment of high-order polynomial bases and Gauss–Lobatto quadrature for rapid convergence on smooth solutions. Applications span wave propagation, elasticity and fluid dynamics, with reduced dispersion errors and minimal mesh refinement requirements compared to low-order schemes.

Numerical and Computational Mathematics publication trend

The graph below shows the total number of articles in numerical and computational mathematics across all publications each year (not limited to Nature Index journals).

Technical terms

Finite-element method: variational discretisation of PDEs using piecewise polynomial basis functions on subdivided domains.

Finite-difference method: approximation of derivatives by difference quotients on structured or unstructured grids.

Spectral method: global approximation using high-order (often orthogonal) basis functions to attain rapid convergence for smooth problems.

Streamline-upwind Petrov-Galerkin stabilisation: a technique adding directional diffusion in convective flows to control oscillations in finite-element solutions.

Isogeometric analysis: a framework employing CAD-based basis functions (e.g. NURBS) to represent geometry and fields with exactness in discretisation.

Shock-capturing scheme: a finite-volume or finite-difference algorithm designed to resolve discontinuities in hyperbolic conservation laws without spurious oscillations.

Virtual element method: a generalisation of FEM allowing arbitrary polygonal/polyhedral elements by using projection operators instead of explicit shape functions.

Norm-resolvent convergence: convergence of inverses (resolvents) of perturbed operators in operator norm, implying stability of spectra under small perturbations.

References

1. An improved method for the calculation of unsaturated–saturated water flow by coupling the FEM and FDM. Scientific Reports (2019).
2. A time-consistent stabilized finite element method for fluids with applications to hemodynamics. Scientific Reports (2023).
3. High-resolution shock-capturing numerical simulations of three-phase immiscible fluids from the unsaturated to the saturated zone. Scientific Reports (2021).
4. The virtual element method. Acta Numerica (2023).
5. Resolvent Convergence for Differential–Difference Operators with Small Variable Translations. Mathematics (2023).
6. A Review: Applications of the Spectral Finite Element Method. Archives of Computational Methods in Engineering (2023).