# Category: Numerical differential equations

Flux limiter
Flux limiters are used in high resolution schemes – numerical schemes used to solve problems in science and engineering, particularly fluid dynamics, described by partial differential equations (PDEs)
Weakened weak form
Weakened weak form (or W2 form) is used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings. These numerical methods are applicable to solid
Milstein method
In mathematics, the Milstein method is a technique for the approximate numerical solution of a stochastic differential equation. It is named after who first published it in 1974.
Leimkuhler–Matthews method
The Leimkuhler-Matthews method (or LM method in its original paper ) is an algorithm for finding discretized solutions to the Brownian dynamics where is a constant and can be thought of as a potential
Cash–Karp method
In numerical analysis, the Cash–Karp method is a method for solving ordinary differential equations (ODEs). It was proposed by Professor Jeff R. Cash from Imperial College London and Alan H. Karp from
Raviart–Thomas basis functions
In applied mathematics, Raviart–Thomas basis functions are vector basis functions used in finite element and boundary element methods. They are regularly used as basis functions when working in electr
Finite difference methods for option pricing
Finite difference methods for option pricing are numerical methods used in mathematical finance for the valuation of options. Finite difference methods were first applied to option pricing by Eduardo
Spectral element method
In the numerical solution of partial differential equations, a topic in mathematics, the spectral element method (SEM) is a formulation of the finite element method (FEM) that uses high degree piecewi
AUSM
AUSM stands for Advection Upstream Splitting Method. It is developed as a numerical inviscid flux function for solving a general system of conservation equations. It is based on the upwind concept and
High-resolution scheme
High-resolution schemes are used in the numerical solution of partial differential equations where high accuracy is required in the presence of shocks or discontinuities. They have the following prope
Numerov's method
Numerov's method (also called Cowell's method) is a numerical method to solve ordinary differential equations of second order in which the first-order term does not appear. It is a fourth-order linear
Variational integrator
Variational integrators are numerical integrators for Hamiltonian systems derived from the Euler–Lagrange equations of a discretized Hamilton's principle. Variational integrators are momentum-preservi
Runge–Kutta method (SDE)
In mathematics of stochastic systems, the Runge–Kutta method is a technique for the approximate numerical solution of a stochastic differential equation. It is a generalisation of the Runge–Kutta meth
Total variation diminishing
In numerical methods, total variation diminishing (TVD) is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The most notable application of this me
Finite-difference time-domain method
Finite-difference time-domain (FDTD) or Yee's method (named after the Chinese American applied mathematician Kane S. Yee, born 1934) is a numerical analysis technique used for modeling computational e
MUSCL scheme
In the study of partial differential equations, the MUSCL scheme is a finite volume method that can provide highly accurate numerical solutions for a given system, even in cases where the solutions ex
Leapfrog integration
In numerical analysis, leapfrog integration is a method for numerically integrating differential equations of the form or equivalently of the formparticularly in the case of a dynamical system of clas
Moving particle semi-implicit method
The moving particle semi-implicit (MPS) method is a computational method for the simulation of incompressible free surface flows. It is a macroscopic, deterministic particle method (Lagrangian mesh-fr
Alternating-direction implicit method
In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester matrix equations. It is a popular method for solving the large matrix equati
Upwind scheme
In computational physics, the term upwind scheme (sometimes advection scheme) typically refers to a class of numerical discretization methods for solving hyperbolic partial differential equations, in
Composite methods for structural dynamics
Composite methods are an approach applied in structural dynamics and related fields. They combine various methods in each time step, in order to acquire the advantages of different methods. The existi
Energy drift
In computer simulations of mechanical systems, energy drift is the gradual change in the total energy of a closed system over time. According to the laws of mechanics, the energy should be a constant
Microscale and macroscale models
Microscale models form a broad class of computational models that simulate fine-scale details, in contrast with macroscale models, which amalgamate details into select categories. Microscale and macro
Mesh generation
Mesh generation is the practice of creating a mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells.Often these cells form a simplicial complex.Usually the
Céa's lemma
Céa's lemma is a lemma in mathematics. Introduced by Jean Céa in his Ph.D. dissertation, it is an important tool for proving error estimates for the finite element method applied to elliptic partial d
Fast marching method
The fast marching method is a numerical method created by James Sethian for solving boundary value problems of the Eikonal equation: Typically, such a problem describes the evolution of a closed surfa
Meshfree methods
In the field of numerical analysis, meshfree methods are those that do not require connection between nodes of the simulation domain, i.e. a mesh, but are rather based on interaction of each node with
Stencil (numerical analysis)
In mathematics, especially the areas of numerical analysis concentrating on the numerical solution of partial differential equations, a stencil is a geometric arrangement of a nodal group that relate
Uniform theory of diffraction
In numerical analysis, the uniform theory of diffraction (UTD) is a high-frequency method for solving electromagnetic scattering problems from electrically small discontinuities or discontinuities in
Discrete Laplace operator
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. For the case of a finite-dimensional graph
Method of moments (electromagnetics)
The method of moments (MoM), also known as the moment method and method of weighted residuals, is a numerical method in computational electromagnetics. It is used in computer programs that simulate th
Runge–Kutta–Fehlberg method
In mathematics, the Runge–Kutta–Fehlberg method (or Fehlberg method) is an algorithm in numerical analysis for the numerical solution of ordinary differential equations. It was developed by the German
Central differencing scheme
In applied mathematics, the central differencing scheme is a finite difference method that optimizes the approximation for the differential operator in the central node of the considered patch and pro
Verlet integration
Verlet integration (French pronunciation: ​[vɛʁˈlɛ]) is a numerical method used to integrate Newton's equations of motion. It is frequently used to calculate trajectories of particles in molecular dyn
Bi-directional delay line
In mathematics, a bi-directional delay line is a numerical analysis technique used in computer simulation for solving ordinary differential equations by converting them to hyperbolic equations. In thi
Numerical dispersion
In computational mathematics, numerical dispersion is a difficulty with computer simulations of continua (such as fluids) wherein the simulated medium exhibits a higher dispersivity than the true medi
Vorticity confinement
Vorticity confinement (VC), a physics-based computational fluid dynamics model analogous to shock capturing methods, was invented by Dr. John Steinhoff, professor at the University of Tennessee Space
Symplectic integrator
In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are cano
WENO methods
In numerical solution of differential equations, WENO (weighted essentially non-oscillatory) methods are classes of high-resolution schemes. WENO are used in the numerical solution of hyperbolic parti
Partial element equivalent circuit
Partial element equivalent circuit method (PEEC) is partial inductance calculation used for interconnect problems from early 1970s which is used for numerical modeling of electromagnetic (EM) properti
Midpoint method
In numerical analysis, a branch of applied mathematics, the midpoint method is a one-step method for numerically solving the differential equation, The explicit midpoint method is given by the formula
Infinite element method
The infinite element method is a numerical method for solving problems of engineering and mathematical physics. It is a modification of finite element method. The method divides the domain concerned i
Multisymplectic integrator
In mathematics, a multisymplectic integrator is a numerical method for the solution of a certain class of partial differential equations, that are said to be multisymplectic. Multisymplectic integrato
Immersed boundary method
In computational fluid dynamics, the immersed boundary method originally referred to an approach developed by Charles Peskin in 1972 to simulate fluid-structure (fiber) interactions. Treating the coup
Weak formulation
Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential e
Characteristic mode analysis
Characteristic modes (CM) form a set of functions which, under specific boundary conditions, diagonalizes operator relating field and induced sources. Under certain conditions, the set of the CM is un
Geometric integrator
In the mathematical field of numerical ordinary differential equations, a geometric integrator is a numerical method that preserves geometric properties of the exact flow of a differential equation.
Lax–Wendroff method
The Lax–Wendroff method, named after Peter Lax and Burton Wendroff, is a numerical method for the solution of hyperbolic partial differential equations, based on finite differences. It is second-order
Extended discrete element method
The extended discrete element method (XDEM) is a numerical technique that extends the dynamics of granular material or particles as described through the classical discrete element method (DEM) (Cunda
Eigenvalues and eigenvectors of the second derivative
Explicit formulas for eigenvalues and eigenvectors of the second derivative with different boundary conditions are provided both for the continuous and discrete cases. In the discrete case, the standa
General linear methods
General linear methods (GLMs) are a large class of numerical methods used to obtain numerical solutions to ordinary differential equations. They include multistage Runge–Kutta methods that use interme
Backward Euler method
In numerical analysis and scientific computing, the backward Euler method (or implicit Euler method) is one of the most basic numerical methods for the solution of ordinary differential equations. It
Momentum (electromagnetic simulator)
Momentum is 3-D planar EM simulation software for electronics and antenna analysis, a partial differential equation solver of Maxwell's equations based on the method of moments. It is a 3-D planar ele
Exponential integrator
Exponential integrators are a class of numerical methods for the solution of ordinary differential equations, specifically initial value problems. This large class of methods from numerical analysis i
Perfectly matched layer
A perfectly matched layer (PML) is an artificial absorbing layer for wave equations, commonly used to truncate computational regions in numerical methods to simulate problems with open boundaries, esp
Numerical resistivity
Numerical resistivity is a problem in computer simulations of ideal magnetohydrodynamics (MHD). It is a form of numerical diffusion. In near-ideal MHD systems, the magnetic field can diffuse only very
Semi-implicit Euler method
In mathematics, the semi-implicit Euler method, also called symplectic Euler, semi-explicit Euler, Euler–Cromer, and Newton–Størmer–Verlet (NSV), is a modification of the Euler method for solving Hami
In mathematics and numerical analysis, an adaptive step size is used in some methods for the numerical solution of ordinary differential equations (including the special case of numerical integration)
Compact stencil
In mathematics, especially in the areas of numerical analysis called numerical partial differential equations, a compact stencil is a type of stencil that uses only nine nodes for its discretization m
List of Runge–Kutta methods
Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation Explicit Runge–Kutta methods take the form Stages for implicit methods of s stages take the more genera
Euler–Maruyama method
In Itô calculus, the Euler–Maruyama method (also called the Euler method) is a method for the approximate numerical solution of a stochastic differential equation (SDE). It is an extension of the Eule
ENO methods
ENO (essentially non-oscillatory) methods are classes of high-resolution schemes in numerical solution of differential equations.
Principles of grid generation
No description available.
Lax–Wendroff theorem
In computational mathematics, the Lax–Wendroff theorem, named after Peter Lax and Burton Wendroff, states that if a conservative numerical scheme for a hyperbolic system of conservation laws converges
Method of fundamental solutions
In scientific computation and simulation, the method of fundamental solutions (MFS) is a technique for solving partial differential equations based on using the fundamental solution as a basis functio
Explicit and implicit methods
Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical approximations to the solutions of time-dependent ordinary and partial differential equations, as is req
Dynamic design analysis method
The dynamic design analysis method (DDAM) is a US Navy-developed analytical procedure for evaluating the design of equipment subject to dynamic loading caused by underwater explosions (UNDEX). The ana
Five-point stencil
In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors". It is us
Fast sweeping method
In applied mathematics, the fast sweeping method is a numerical method for solving boundary value problems of the Eikonal equation. where is an open set in , is a function with positive values, is a w
Spectral method
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differenti
Finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the tradi
Split-step method
In numerical analysis, the split-step (Fourier) method is a pseudo-spectral numerical method used to solve nonlinear partial differential equations like the nonlinear Schrödinger equation. The name ar
Rayleigh–Ritz method
The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz
Finite difference method
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial dom
Linear multistep method
Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in ti
Volume of fluid method
In computational fluid dynamics, the volume of fluid (VOF) method is a free-surface modelling technique, i.e. a numerical technique for tracking and locating the free surface (or fluid–fluid interface
Strang splitting
Strang splitting is a numerical method for solving differential equations that are decomposable into a sum of differential operators. It is named after Gilbert Strang. It is used to speed up calculati
Runge–Kutta methods
In numerical analysis, the Runge–Kutta methods (English: /ˈrʊŋəˈkʊtɑː/ RUUNG-ə-KUUT-tah) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discr
Higher-order compact finite difference scheme
High-order compact finite difference schemes are used for solving third-order differential equations created during the study of obstacle boundary value problems. They have been shown to be highly acc
Shooting method
In numerical analysis, the shooting method is a method for solving a boundary value problem by reducing it to an initial value problem. It involves finding solutions to the initial value problem for d
Particle-in-cell
In plasma physics, the particle-in-cell (PIC) method refers to a technique used to solve a certain class of partial differential equations. In this method, individual particles (or fluid elements) in
Singular boundary method
In numerical analysis, the singular boundary method (SBM) belongs to a family of meshless boundary collocation techniques which include the method of fundamental solutions (MFS), boundary knot method
Euler method
In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initi
Material point method
The material point method (MPM) is a numerical technique used to simulate the behavior of solids, liquids, gases, and any other continuum material. Especially, it is a robust spatial discretization me
Numerical methods for ordinary differential equations
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerica
Diffuse element method
In numerical analysis the diffuse element method (DEM) or simply diffuse approximation is a meshfree method. The diffuse element method was developed by B. Nayroles, G. Touzot and Pierre Villon at the
Finite difference
A finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by fini
Courant–Friedrichs–Lewy condition
In mathematics, the convergence condition by Courant–Friedrichs–Lewy is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically
Non-compact stencil
In numerical mathematics, a non-compact stencil is a type of discretization method, where any node surrounding the node of interest may be used in the calculation. Its computational time grows with an
Numerical diffusion
Numerical diffusion is a difficulty with computer simulations of continua (such as fluids) wherein the simulated medium exhibits a higher diffusivity than the true medium. This phenomenon can be parti
Discrete Poisson equation
In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator. The discrete Poisso
Finite element method in structural mechanics
The finite element method (FEM) is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems
Zero stability
Zero-stability, also known as D-stability in honor of Germund Dahlquist, refers to the stability of a numerical scheme applied to the simple initial value problem . A linear multistep method is zero-s
Modal analysis using FEM
The goal of modal analysis in structural mechanics is to determine the natural mode shapes and frequencies of an object or structure during free vibration. It is common to use the finite element metho
In numerical analysis, adaptive mesh refinement (AMR) is a method of adapting the accuracy of a solution within certain sensitive or turbulent regions of simulation, dynamically and during the time th
MacCormack method
In computational fluid dynamics, the MacCormack method is a widely used discretization scheme for the numerical solution of hyperbolic partial differential equations. This second-order finite differen
Analytic element method
The analytic element method (AEM) is a numerical method used for the solution of partial differential equations. It was initially developed by O.D.L. Strack at the University of Minnesota. It is simil
Multiphase particle-in-cell method
The multiphase particle-in-cell method (MP-PIC) is a numerical method for modeling particle-fluid and particle-particle interactions in a computational fluid dynamics (CFD) calculation. The MP-PIC met
Stiff equation
In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small.
Direct stiffness method
As one of the methods of structural analysis, the direct stiffness method, also known as the matrix stiffness method, is particularly suited for computer-automated analysis of complex structures inclu
Constraint (computational chemistry)
In computational chemistry, a constraint algorithm is a method for satisfying the Newtonian motion of a rigid body which consists of mass points. A restraint algorithm is used to ensure that the dista
Godunov's theorem
In numerical analysis and computational fluid dynamics, Godunov's theorem — also known as Godunov's order barrier theorem — is a mathematical theorem important in the development of the theory of high
Lax–Friedrichs method
The Lax–Friedrichs method, named after Peter Lax and Kurt O. Friedrichs, is a numerical method for the solution of hyperbolic partial differential equations based on finite differences. The method can
Cyclic reduction
Cyclic reduction is a numerical method for solving large linear systems by repeatedly splitting the problem. Each step eliminates even or odd rows and columns of a matrix and remains in a similar form
Cell lists
Cell lists (also sometimes referred to as cell linked-lists) is a data structure in molecular dynamics simulations to find all atom pairs within a given cut-off distance of each other. These pairs are
Godunov's scheme
In numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by S. K. Godunov in 1959, for solving partial differential equations. One can thi
Discrete calculus
Discrete calculus or the calculus of discrete functions, is the mathematical study of incremental change, in the same way that geometry is the study of shape and algebra is the study of generalization
Natural element method
The natural element method (NEM) is a meshless method to solve partial differential equation, where the elements do not have a predefined shape as in the finite element method, but depend on the geome
Backward differentiation formula
The backward differentiation formula (BDF) is a family of implicit methods for the numerical integration of ordinary differential equations. They are linear multistep methods that, for a given functio
Boundary knot method
In numerical mathematics, the boundary knot method (BKM) is proposed as an alternative boundary-type meshfree distance function collocation scheme. Recent decades have witnessed a research boom on the
Infinite difference method
In mathematics, infinite difference methods are numerical methods for solving differential equations by approximating them with difference equations, in which infinite differences approximate the deri
Multilevel fast multipole method
The multilevel fast multipole method (MLFMM) is used along with method of moments (MoM) a numerical computational method of solving linear partial differential equations which have been formulated as
Numerical methods for partial differential equations
Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs). In principle, specialized methods
Walk-on-spheres method
In mathematics, the walk-on-spheres method (WoS) is a numerical probabilistic algorithm, or Monte-Carlo method, used mainly in order to approximate the solutions of some specific boundary value proble
History of numerical solution of differential equations using computers
Differential equations, in particular Euler equations, rose in prominence during World War II in calculating the accurate trajectory of ballistics, both rocket-propelled and gun or cannon type project
Extended finite element method
The extended finite element method (XFEM), is a numerical technique based on the generalized finite element method (GFEM) and the partition of unity method (PUM). It extends the classical finite eleme
Parareal
Parareal is a parallel algorithm from numerical analysis and used for the solution of initial value problems.It was introduced in 2001 by Lions, Maday and Turinici. Since then, it has become one of th
Mimesis (mathematics)
In mathematics, mimesis is the quality of a numerical method which imitates some properties of the continuum problem. The goal of numerical analysis is to approximate the continuum, so instead of solv
Heun's method
In mathematics and computational science, Heun's method may refer to the improved or modified Euler's method (that is, the explicit trapezoidal rule), or a similar two-stage Runge–Kutta method. It is
Beeman's algorithm
Beeman's algorithm is a method for numerically integrating ordinary differential equations of order 2, more specifically Newton's equations of motion . It was designed to allow high numbers of particl
Smoothed finite element method
Smoothed finite element methods (S-FEM) are a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining meshfree methods with the fini
Direct multiple shooting method
In the area of mathematics known as numerical ordinary differential equations, the direct multiple shooting method is a numerical method for the solution of boundary value problems. The method divides
Boundary element method
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in boundary integral form)
Shock-capturing method
In computational fluid dynamics, shock-capturing methods are a class of techniques for computing inviscid flows with shock waves. The computation of flow containing shock waves is an extremely difficu
Beam and Warming scheme
In numerical mathematics, Beam and Warming scheme or Beam–Warming implicit scheme introduced in 1978 by Richard M. Beam and R. F. Warming, is a second order accurate implicit scheme, mainly used for s
Dormand–Prince method
In numerical analysis, the Dormand–Prince (RKDP) method or DOPRI method, is an embedded method for solving ordinary differential equations. The method is a member of the Runge–Kutta family of ODE solv
Galerkin method
In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commo
Finite-difference frequency-domain method
The finite-difference frequency-domain (FDFD) method is a numerical solution method for problems usually in electromagnetism and sometimes in acoustics, based on finite-difference approximations of th
Discontinuous Galerkin method
In applied mathematics, discontinuous Galerkin methods (DG methods) form a class of numerical methods for solving differential equations. They combine features of the finite element and the finite vol
Method of lines
The method of lines (MOL, NMOL, NUMOL) is a technique for solving partial differential equations (PDEs) in which all but one dimension is discretized. By reducing a PDE to a single continuous dimensio
Roe solver
The Roe approximate Riemann solver, devised by Phil Roe, is an approximate Riemann solver based on the Godunov scheme and involves finding an estimate for the intercell numerical flux or Godunov flux
Discrete element method
A discrete element method (DEM), also called a distinct element method, is any of a family of numerical methods for computing the motion and effect of a large number of small particles. Though DEM is
Pantelides algorithm
Pantelides algorithm in mathematics is a systematic method for reducing high-index systems of differential-algebraic equations to lower index. This is accomplished by selectively adding differentiated
Loubignac iteration
In applied mathematics, Loubignac iteration is an iterative method in finite element methods. It gives continuous stress field. It is named after Gilles Loubignac, who published the method in 1977.
Finite volume method
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations.In the finite volume method, volume integrals in a partial
Particle method
In the field of numerical analysis, particle methods discretize fluid into particles. Particle methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computi
In numerical mathematics, the gradient discretisation method (GDM) is a framework which contains classical and recent numerical schemes for diffusion problems of various kinds: linear or non-linear, s
Finite pointset method
In applied mathematics, the name finite pointset method is a general approach for the numerical solution of problems in continuum mechanics, such as the simulation of fluid flows. In this approach (of
Laplacian matrix
In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Name
Upwind differencing scheme for convection
The upwind differencing scheme is a method used in numerical methods in computational fluid dynamics for convection–diffusion problems. This scheme is specific for Peclet number greater than 2 or less
Kronecker sum of discrete Laplacians
In mathematics, the Kronecker sum of discrete Laplacians, named after Leopold Kronecker, is a discrete version of the separation of variables for the continuous Laplacian in a rectangular cuboid domai
Crank–Nicolson method
In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in
Numerical methods for differential equations
Numerical methods for differential equations may refer to: * Numerical methods for ordinary differential equations, methods used to find numerical approximations to the solutions of ordinary differen
Boundary particle method
In applied mathematics, the boundary particle method (BPM) is a boundary-only meshless (meshfree) collocation technique, in the sense that none of inner nodes are required in the numerical solution of
Fast multipole method
The fast multipole method (FMM) is a numerical technique that was developed to speed up the calculation of long-ranged forces in the n-body problem. It does this by expanding the system Green's functi
False diffusion
False diffusion is a type of error observed when the upwind scheme is used to approximate the convection term in convection–diffusion equations. The more accurate central difference scheme can be used
FTCS scheme
In numerical analysis, the FTCS (Forward Time Centered Space) method is a finite difference method used for numerically solving the heat equation and similar parabolic partial differential equations.
Stochastic Eulerian Lagrangian method
In computational fluid dynamics, the Stochastic Eulerian Lagrangian Method (SELM) is an approach to capture essential features of fluid-structure interactions subject to thermal fluctuations while int
Newmark-beta method
The Newmark-beta method is a method of numerical integration used to solve certain differential equations. It is widely used in numerical evaluation of the dynamic response of structures and solids su
Finite difference coefficient
In mathematics, to approximate a derivative to an arbitrary order of accuracy, it is possible to use the finite difference. A finite difference can be central, forward or backward.
Interval boundary element method
Interval boundary element method is classical boundary element method with the interval parameters.Boundary element method is based on the following integral equation The exact interval solution on th
Smoothed-particle hydrodynamics
Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan
Collocation method
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite-di
Virtual Cell
Virtual Cell (VCell) is an open-source software platform for modeling and simulation of living organisms, primarily cells. It has been designed to be a tool for a wide range of scientists, from experi
Regularized meshless method
In numerical mathematics, the regularized meshless method (RMM), also known as the singular meshless method or desingularized meshless method, is a meshless boundary collocation method designed to sol
Lax equivalence theorem
In numerical analysis, the Lax equivalence theorem is a fundamental theorem in the analysis of finite difference methods for the numerical solution of partial differential equations. It states that fo
Proper orthogonal decomposition
The proper orthogonal decomposition is a numerical method that enables a reduction in the complexity of computer intensive simulations such as computational fluid dynamics and structural analysis (lik
Coupled mode theory
Coupled mode theory (CMT) is a perturbational approach for analyzing the coupling of vibrational systems (mechanical, optical, electrical, etc.) in space or in time. Coupled mode theory allows a wide
Stretched grid method
The stretched grid method (SGM) is a numerical technique for finding approximate solutions of various mathematical and engineering problems that can be related to an elastic grid behavior.In particula
Trefftz method
In mathematics, the Trefftz method is a method for the numerical solution of partial differential equations named after the German mathematician (de) (1888–1937). It falls within the class of finite e
L-stability
Within mathematics regarding differential equations, L-stability is a special case of A-stability, a property of Runge–Kutta methods for solving ordinary differential equations.A method is L-stable if
P-FEM
p-FEM or the p-version of the finite element method is a numerical method for solving partial differential equations. It is a discretization strategy in which the finite element mesh is fixed and the
Quantized state systems method
The quantized state systems (QSS) methods are a family of numerical integration solvers based on the idea of state quantization, dual to the traditional idea of time discretization.Unlike traditional