mathematics

Numerical methods for partial differential equations

Numerical methods for partial differential equations is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Numerical methods for partial differential equations rather than just read about it. In short: 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 for hyperbolic, parabolic or elliptic partial differential equations exist.

Key takeaways

Numerical methods for partial differential equations belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Numerical methods for partial differential equations to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Numerical methods for partial differential equations from memory before moving on to harder problems.

Reference excerpt

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 for hyperbolic, parabolic or elliptic partial differential equations exist.

Overview of methods

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

The method of lines (MOL, NMOL, NUMOL) 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 equations (ODEs) and differential 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 as open source resources. 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.

Finite element method

The finite element method (FEM) is a numerical technique for finding approximate solutions to boundary value problems 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.

Gradient discretization method

The gradient discretization method (GDM) is a 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

The finite-volume method is a numerical technique for representing and evaluating partial differential equations 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 integrals, 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 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

Spectral methods are techniques used in applied mathematics and scientific computing to numerically solve certain differential equations, 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 sinusoids) and then to choose the coefficients in the sum that best satisfy the differential equation. Spectral methods and finite 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 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. However, there are no known three-dimensional single domain spectral shock capturing results. 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

Meshfree methods do not require a mesh connecting the data points of the simulation domain. 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

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Worked examples

Example 1 — a first encounter with Numerical methods for partial differential equations

Start with the simplest possible case. Write down what Numerical methods for partial differential equations claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Numerical methods for partial differential equations before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Numerical methods for partial differential equations ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Numerical methods for partial differential equations

In research: Numerical methods for partial differential equations appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Numerical methods for partial differential equations in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Numerical methods for partial differential equations is common in secondary-school and first-year university syllabi. It links to neighbouring topics Numerical differential equations, Partial differential equations, so understanding it makes those chapters shorter.
In everyday life: Look for Numerical methods for partial differential equations outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Numerical methods for partial differential equations in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Numerical methods for partial differential equations means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Numerical methods for partial differential equations out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Numerical methods for partial differential equations in simple terms?

Why does Numerical methods for partial differential equations matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Numerical methods for partial differential equations?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Numerical methods for partial differential equations.