View source for Numerical methods for partial differential equations

{{Short description|Branch of numerical analysis}}
{{for|the academic journal|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).<ref>{{Cite book|last=Pinder|first=George F.|url=https://www.worldcat.org/oclc/1015215158|title=Numerical methods for solving partial differential equations : a comprehensive introduction for scientists and engineers|date=2018|isbn=978-1-119-31636-7|location=Hoboken, NJ|oclc=1015215158}}</ref><ref>{{Citation|title=Numerical methods|date=2005|url=https://www.cambridge.org/core/books/an-introduction-to-partial-differential-equations/numerical-methods/84C48DE040E38F7098D13876F7FE02DE|work=An Introduction to Partial Differential Equations|pages=309–336|editor-last=Rubinstein|editor-first=Jacob|place=Cambridge|publisher=Cambridge University Press|doi=10.1017/cbo9780511801228.012|isbn=978-0-511-80122-8|access-date=2021-11-15|editor2-last=Pinchover|editor2-first=Yehuda|url-access=subscription}}</ref>

In principle, specialized methods for [[Hyperbolic partial differential equation|hyperbolic]],<ref>{{Cite web|title=Hyperbolic partial differential equation, numerical methods - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Hyperbolic_partial_differential_equation,_numerical_methods|access-date=2021-11-15|website=encyclopediaofmath.org}}</ref> [[Parabolic partial differential equation|parabolic]]<ref>{{Cite web|title=Parabolic partial differential equation, numerical methods - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Parabolic_partial_differential_equation,_numerical_methods|access-date=2021-11-15|website=encyclopediaofmath.org}}</ref> or [[Elliptic partial differential equation|elliptic partial differential equations]]<ref>{{Cite web|title=Elliptic partial differential equation, numerical methods - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Elliptic_partial_differential_equation,_numerical_methods|access-date=2021-11-15|website=encyclopediaofmath.org}}</ref> exist.<ref>{{Cite book|last=Evans|first=Gwynne|url=https://www.worldcat.org/oclc/41572731|title=Numerical methods for partial differential equations|date=2000|publisher=Springer|others=J. M. Blackledge, P. Yardley|isbn=3-540-76125-X|location=London|oclc=41572731}}</ref><ref>{{Cite book|last=Grossmann|first=Christian|url=https://www.worldcat.org/oclc/191468303|title=Numerical treatment of partial differential equations|date=2007|publisher=Springer|others=Hans-Görg Roos, M. Stynes|isbn=978-3-540-71584-9|location=Berlin|oclc=191468303}}</ref>

==Overview of methods==

===Finite difference method===
{{Main|Finite difference method}}
In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.

===Method of lines===
{{Main|Method of lines}}
The '''method of lines''' (MOL, NMOL, NUMOL<ref name=Schiesser1991>{{cite book|last=Schiesser|first=W. E.|year=1991|title=The Numerical Method of Lines|publisher=Academic Press|isbn=0-12-624130-9|url-access=registration|url=https://archive.org/details/numericalmethodo0000schi}}</ref><ref name=Hamdi2007>Hamdi, S., W. E. Schiesser and G. W. Griffiths (2007), [http://www.scholarpedia.org/article/Method_of_Lines  Method of lines], ''Scholarpedia'', 2(7):2859.</ref><ref name=Schiesser2009>{{cite book|last1=Schiesser|first1=W. E. |first2=G. W. |last2=Griffiths|year=2009|title=A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab|publisher=Cambridge University Press|isbn=978-0-521-51986-1}}</ref>) is a technique for solving [[partial differential equations]] (PDEs) in which all dimensions except one are discretized.  MOL allows standard, general-purpose methods and software, developed for the numerical integration of [[ordinary differential equation]]s (ODEs) and [[differential algebraic equation]]s (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 as [[open source]] resources.<ref name=Lee2004>{{cite book|last1=Lee |first1=H. J. |first2=W. E. |last2=Schiesser |year=2004|title=Ordinary and Partial Differential Equation Routines in C, C++, Fortran, Java, Maple and Matlab|publisher=CRC Press|isbn=1-58488-423-1}}</ref>

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.<ref name=Sarmin1962>E. N. Sarmin, L. A. Chudov (1963), On the stability of the numerical integration of systems of ordinary differential equations arising in the use of the straight line method, ''USSR Computational Mathematics and Mathematical Physics'', '''3'''(6), (1537–1543).</ref>

===Finite element method===
{{Main|Finite element method}}
The '''finite element method (FEM)''' is a [[numerical analysis|numerical technique]] for finding approximate solutions to [[boundary value problem]]s for [[differential equations]]. It uses [[variational methods]] (the [[calculus 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 larger [[domain of a function|domain]].

===Gradient discretization method===
{{Main|Gradient discretization method}}
The '''gradient discretization method (GDM)''' is a [[numerical analysis|numerical 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.

===Finite volume method===
{{Main|Finite volume method}}
The '''finite-volume method''' is a [[numerical analysis|numerical technique]] for representing and evaluating [[partial differential equation]]s in the form of algebraic equations [LeVeque, 2002; Toro, 1999].  
Similar to the [[finite difference method]] or [[finite 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 a [[divergence]] term are converted to [[surface integral]]s, using the [[divergence 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 are [[Conservation law (physics)|conservative]]. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many [[computational fluid dynamics]] packages.

===Spectral method===
{{Main|Spectral method}}
'''Spectral methods''' are techniques used in [[applied mathematics]] and [[scientific computing]] to numerically solve certain [[differential equation]]s, often involving the use of the [[fast Fourier transform]]. The idea is to write the solution of the differential equation as a sum of certain "basis functions" (for example, as a [[Fourier series]], which is a sum of [[Sine wave|sinusoid]]s) and then to choose the coefficients in the sum that best satisfy the differential equation.

Spectral methods and [[finite element method]]s 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 a ''global approach'' while finite element methods use a ''local approach''. Partially for this reason, spectral methods have excellent error properties, with the so-called "exponential convergence" being the fastest possible, when the solution is [[Smooth function|smooth]]. However, there are no known three-dimensional single domain spectral [[shock capturing]] results.<ref name="CHQZ">[https://books.google.com/books?id=7COgEw5_EBQC pp 235, Spectral Methods]: evolution to complex geometries and applications to fluid dynamics, By Canuto, Hussaini, Quarteroni and Zang, Springer, 2007.</ref> In the finite element community, a method where the degree of the elements is very high or increases as the grid parameter ''h'' decreases to zero is sometimes called a [[spectral element method]].

===Meshfree methods===
{{Main|Meshfree methods}}
'''Meshfree methods''' do not require a mesh connecting the data points of the simulation domain.<ref>{{Cite journal |last1=Chen |first1=Shang-Ying |last2=Wei |first2=Jian-Yu |last3=Hsu |first3=Kuo-Chin |date=2023-10-01 |title=Data assimilation for real-time subsurface flow modeling with dynamically adaptive meshless node adjustments |url=https://doi.org/10.1007/s00366-023-01897-6 |journal=Engineering with Computers |volume=40 |issue=3 |pages=1893–1925 |language=en |doi=10.1007/s00366-023-01897-6 |issn=1435-5663|url-access=subscription }}</ref> 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===
{{Main|Domain decomposition method}}
'''Domain decomposition methods''' solve a [[boundary value problem]] by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains. A [[coarse 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 for [[parallel computing]]. Domain decomposition methods are typically used as [[preconditioner]]s for [[Krylov space]] [[iterative method]]s, such as the [[conjugate gradient method]] or [[GMRES]].

In overlapping domain decomposition methods, the subdomains overlap by more than the interface. Overlapping domain decomposition methods include the [[Schwarz alternating method]] and the [[additive Schwarz method]]. Many domain decomposition methods can be written and analyzed as a special case of the [[abstract additive Schwarz method]].

In non-overlapping methods, the subdomains intersect only on their interface. In primal methods, such as [[Balancing domain decomposition]] and [[BDDC]], 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 as [[FETI]], the continuity of the solution across the subdomain interface is enforced by [[Lagrange multiplier]]s. The [[FETI-DP]] method is hybrid between a dual and a primal method.

Non-overlapping domain decomposition methods are also called '''iterative substructuring methods'''.

[[Mortar method]]s 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 by [[multiple-point constraint]]s.

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 methods===
{{Main|Multigrid method}}
'''Multigrid (MG) methods''' in [[numerical analysis]] are a group of [[algorithm]]s for solving [[differential equations]] using a [[hierarchy]] of [[discretization]]s. They are an example of a class of techniques called [[Multiresolution analysis|multiresolution methods]], very useful in (but not limited to) problems exhibiting [[Multiscale modeling|multiple scales]] of behavior. For example, many basic [[relaxation method]]s exhibit different rates of convergence for short- and long-wavelength components, suggesting these different scales be treated differently, as in a [[Fourier analysis]] approach to multigrid.<ref>{{cite book |title=Practical Fourier analysis for multigrid methods |author1=Roman Wienands |author2=Wolfgang Joppich |page=17 |url=https://books.google.com/books?id=IOSux5GxacsC&pg=PA17 |isbn=1-58488-492-4 |publisher=CRC Press |year=2005}}</ref> MG methods can be used as solvers as well as [[preconditioner]]s.

The main idea of multigrid is to accelerate the convergence of a basic iterative method by ''global'' correction from time to time, accomplished by solving a [[coarse problem]]. This principle is similar to [[interpolation]] between coarser and finer grids. The typical application for multigrid is in the numerical solution of [[elliptic partial differential equation]]s in two or more dimensions.<ref>{{cite book |title=Multigrid |author1=U. Trottenberg |author2=C. W. Oosterlee |author3=A. Schüller |publisher=Academic Press |year=2001 |isbn=0-12-701070-X |url=https://books.google.com/books?id=-og1wD-Nx_wC&q=elliptic}}</ref>

Multigrid methods can be applied in combination with any of the common discretization techniques. For example, the [[finite element method]] may be recast as a multigrid method.<ref>{{cite book |title=Multigrid finite element methods for electromagnetic field modeling |author1=Yu Zhu |author2=Andreas C. Cangellaris |url=https://books.google.com/books?id=amq9j71_nqAC&pg=PA132 |page=132 ''ff'' |isbn=0-471-74110-8 |year=2006 |publisher=Wiley}}</ref> 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 and [[boundary condition]]s. They do not depend on the [[Separable partial differential equation|separability 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 the [[Lamé system]] of [[Elasticity (physics)|elasticity]] or the [[Navier–Stokes equations]].<ref>{{cite thesis |last=Shah |first=Tasneem Mohammad |title=Analysis of the multigrid method |bibcode=1989STIN...9123418S |publisher=Oxford University |year=1989 }}</ref>

==Comparison==
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 in [[engineering]] and in [[computational 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.

==See also==
* {{slink|List of numerical analysis topics#Numerical methods for partial differential equations}}
* [[Numerical methods for ordinary differential equations]]

==Further reading==
*{{Cite book|last=LeVeque|first=Randall J.|url=http://link.springer.com/10.1007/978-3-0348-8629-1|title=Numerical Methods for Conservation Laws|date=1992|publisher=Birkhäuser Basel|isbn=9783764327231|location=Basel|doi=10.1007/978-3-0348-8629-1 |accessdate=2021-11-15}}
*{{Cite book|last1=Anderson|first1=Dale A.|title=Computational fluid mechanics and heat transfer|last2=Pletcher|first2=Richard H.|last3=Tannehill|first3=John C.|date=2013|publisher=CRC Press, Taylor & Francis Group|isbn=9781591690375|edition=3rd.|series=Series in computational and physical processes in mechanics and thermal sciences|location=Boca Raton}}

==References==
<references />

==External links==
*[http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/ Numerical Methods for Partial Differential Equations] course at [[MIT OpenCourseWare]].
*[https://web.archive.org/web/20060118052359/http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/ IMS], the Open Source IMTEK Mathematica Supplement (IMS)
*[http://www3.nd.edu/~dbalsara/Numerical-PDE-Course/ Numerical PDE Techniques for Scientists and Engineers], open access Lectures and Codes for Numerical PDEs

{{Numerical PDE}}
{{Industrial and applied mathematics}}
{{DEFAULTSORT:Numerical Partial Differential Equations}}
[[Category:Numerical differential equations| ]]
[[Category:Partial differential equations]]