Numerical methods for partial differential equations is the branch ofnumerical analysis that studies the numerical solution ofpartial differential equations (PDEs).[1][2]
In principle, specialized methods forhyperbolic,[3]parabolic[4] orelliptic partial differential equations[5] exist.[6][7]
In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.
Themethod of lines (MOL, NMOL, NUMOL[8][9][10]) is a technique for solvingpartial differential equations (PDEs) in which all dimensions except one are discretized. MOL allows standard, general-purpose methods and software, developed for the numerical integration ofordinary differential equations (ODEs) anddifferential algebraic equations (DAEs), to be used. A large number of integration routines have been developed over the years in many different programming languages, and some have been published asopen source resources.[11]
The method of lines most often refers to the construction or analysis of numerical methods for partial differential equations that proceeds by first discretizing the spatial derivatives only and leaving the time variable continuous. This leads to a system of ordinary differential equations to which a numerical method for initial value ordinary equations can be applied. The method of lines in this context dates back to at least the early 1960s.[12]
Thefinite element method (FEM) is anumerical technique for finding approximate solutions toboundary value problems fordifferential equations. It usesvariational methods (thecalculus of variations) to minimize an error function and produce a stable solution. Analogous to the idea that connecting many tiny straight lines can approximate a larger circle, FEM encompasses all the methods for connecting many simple element equations over many small subdomains, named finite elements, to approximate a more complex equation over a largerdomain.
Thegradient discretization method (GDM) is anumerical technique that encompasses a few standard or recent methods. It is based on the separate approximation of a function and of its gradient. Core properties allow the convergence of the method for a series of linear and nonlinear problems, and therefore all the methods that enter the GDM framework (conforming and nonconforming finite element, mixed finite element, mimetic finite difference...) inherit these convergence properties.
Thefinite-volume method is anumerical technique for representing and evaluatingpartial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]. Similar to thefinite difference method orfinite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a partial differential equation that contain adivergence term are converted tosurface integrals, using thedivergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods areconservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in manycomputational fluid dynamics packages.
Spectral methods are techniques used inapplied mathematics andscientific computing to numerically solve certaindifferential equations, often involving the use of thefast Fourier transform. The idea is to write the solution of the differential equation as a sum of certain "basis functions" (for example, as aFourier series, which is a sum ofsinusoids) and then to choose the coefficients in the sum that best satisfy the differential equation.
Spectral methods andfinite element methods are closely related and built on the same ideas; the main difference between them is that spectral methods use basis functions that are nonzero over the whole domain, while finite element methods use basis functions that are nonzero only on small subdomains. In other words, spectral methods take on aglobal approach while finite element methods use alocal approach. Partially for this reason, spectral methods have excellent error properties, with the so-called "exponential convergence" being the fastest possible, when the solution issmooth. However, there are no known three-dimensional single domain spectralshock capturing results.[13] In the finite element community, a method where the degree of the elements is very high or increases as the grid parameterh decreases to zero is sometimes called aspectral element method.
Meshfree methods do not require a mesh connecting the data points of the simulation domain.[14] Meshfree methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computing time and programming effort.
Domain decomposition methods solve aboundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains. Acoarse problem with one or few unknowns per subdomain is used to further coordinate the solution between the subdomains globally. The problems on the subdomains are independent, which makes domain decomposition methods suitable forparallel computing. Domain decomposition methods are typically used aspreconditioners forKrylov spaceiterative methods, such as theconjugate gradient method orGMRES.
In overlapping domain decomposition methods, the subdomains overlap by more than the interface. Overlapping domain decomposition methods include theSchwarz alternating method and theadditive Schwarz method. Many domain decomposition methods can be written and analyzed as a special case of theabstract additive Schwarz method.
In non-overlapping methods, the subdomains intersect only on their interface. In primal methods, such asBalancing domain decomposition andBDDC, the continuity of the solution across subdomain interface is enforced by representing the value of the solution on all neighboring subdomains by the same unknown. In dual methods, such asFETI, the continuity of the solution across the subdomain interface is enforced byLagrange multipliers. TheFETI-DP method is hybrid between a dual and a primal method.
Non-overlapping domain decomposition methods are also callediterative substructuring methods.
Mortar methods are discretization methods for partial differential equations, which use separate discretization on nonoverlapping subdomains. The meshes on the subdomains do not match on the interface, and the equality of the solution is enforced by Lagrange multipliers, judiciously chosen to preserve the accuracy of the solution. In the engineering practice in the finite element method, continuity of solutions between non-matching subdomains is implemented bymultiple-point constraints.
Finite element simulations of moderate size models require solving linear systems with millions of unknowns. Several hours per time step is an average sequential run time, therefore, parallel computing is a necessity. Domain decomposition methods embody large potential for a parallelization of the finite element methods, and serve a basis for distributed, parallel computations.
Multigrid (MG) methods innumerical analysis are a group ofalgorithms for solvingdifferential equations using ahierarchy ofdiscretizations. They are an example of a class of techniques calledmultiresolution methods, very useful in (but not limited to) problems exhibitingmultiple scales of behavior. For example, many basicrelaxation methods exhibit different rates of convergence for short- and long-wavelength components, suggesting these different scales be treated differently, as in aFourier analysis approach to multigrid.[15] MG methods can be used as solvers as well aspreconditioners.
The main idea of multigrid is to accelerate the convergence of a basic iterative method byglobal correction from time to time, accomplished by solving acoarse problem. This principle is similar tointerpolation between coarser and finer grids. The typical application for multigrid is in the numerical solution ofelliptic partial differential equations in two or more dimensions.[16]
Multigrid methods can be applied in combination with any of the common discretization techniques. For example, thefinite element method may be recast as a multigrid method.[17] In these cases, multigrid methods are among the fastest solution techniques known today. In contrast to other methods, multigrid methods are general in that they can treat arbitrary regions andboundary conditions. They do not depend on theseparability of the equations or other special properties of the equation. They have also been widely used for more-complicated non-symmetric and nonlinear systems of equations, like theLamé system ofelasticity or theNavier–Stokes equations.[18]
The finite difference method is often regarded as the simplest method to learn and use. The finite element and finite volume methods are widely used inengineering and incomputational fluid dynamics, and are well suited to problems in complicated geometries.Spectral methods are generally the most accurate, provided that the solutions are sufficiently smooth.
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