Innumerical analysis,finite-difference methods (FDM) are a class of numerical techniques for solvingdifferential equations by approximatingderivatives withfinite differences. Both the spatial domain and time domain (if applicable) arediscretized, or broken into a finite number of intervals, and the values of the solution at the end points of the intervals are approximated by solving algebraic equations containing finite differences and values from nearby points.

Finite difference methods convertordinary differential equations (ODE) orpartial differential equations (PDE), which may benonlinear, into asystem of linear equations that can be solved bymatrix algebra techniques. Modern computers can perform theselinear algebra computations efficiently, and this, along with their relative ease of implementation, has led to the widespread use of FDM in modern numerical analysis.^[1]Today, FDMs are one of the most common approaches to the numerical solution of PDE, along withfinite element methods.^[1]

For an-times differentiable function, byTaylor's theorem theTaylor series expansion is given as$${\displaystyle f(x_0+h)=f(x_0)+\frac{f^{\prime }(x_0)}{1!}h+\frac{f^{(2)}(x_0)}{2!}{h}^2+\cdots +\frac{f^{(n)}(x_0)}{n!}{h}^n+R_n(x),}$$

Wheren! denotes thefactorial ofn, andR_n(x) is a remainder term, denoting the difference between the Taylor polynomial of degreen and the original function.

Following is the process to derive an approximation for the first derivative of the functionf by first truncating the Taylor polynomial plus remainder:$${\displaystyle f(x_0+h)=f(x_0)+f^{\prime }(x_0)h+R_1(x).}$$Dividing across byh gives:$${\displaystyle \frac{f(x_0+h)}{h}=\frac{f(x_0)}{h}+f^{\prime }(x_0)+\frac{R_1(x)}{h}}$$Solving for${\displaystyle f^{\prime }(x_0)}$:$${\displaystyle f^{\prime }(x_0)=\frac{f(x_0+h)-f(x_0)}{h}-\frac{R_1(x)}{h}.}$$

Assuming that${\displaystyle R_1(x)}$ is sufficiently small, the approximation of the first derivative off is:$${\displaystyle f^{\prime }(x_0)\approx \frac{f(x_0+h)-f(x_0)}{h}.}$$

This is similar to the definition of derivative, which is:$${\displaystyle f^{\prime }(x_0)=\underset{h\to 0}{lim}\frac{f(x_0+h)-f(x_0)}{h}.}$$except for the limit towards zero (the method is named after this).

The error in a method's solution is defined as the difference between the approximation and the exact analytical solution. The two sources of error in finite difference methods areround-off error, the loss of precision due to computer rounding of decimal quantities, andtruncation error ordiscretization error, the difference between the exact solution of the original differential equation and the exact quantity assuming perfect arithmetic (no round-off).

To use a finite difference method to approximate the solution to a problem, one must first discretize the problem's domain. This is usually done by dividing the domain into a uniform grid (see image). This means that finite-difference methods produce sets of discrete numerical approximations to the derivative, often in a "time-stepping" manner.

An expression of general interest is thelocal truncation error of a method. Typically expressed usingBig-O notation, local truncation error refers to the error from a single application of a method. That is, it is the quantity${\displaystyle f^{\prime }(x_i)-f_i^{\prime }}$ if${\displaystyle f^{\prime }(x_i)}$ refers to the exact value and${\displaystyle f_i^{\prime }}$ to the numerical approximation. The remainder term of the Taylor polynomial can be used to analyzelocal truncation error. Using theLagrange form of the remainder from the Taylor polynomial for${\displaystyle f(x_0+h)}$, which isthe dominant term of the local truncation error can be discovered. For example, again using the forward-difference formula for the first derivative, knowing that${\displaystyle f(x_i)=f(x_0+ih)}$,$${\displaystyle f(x_0+ih)=f(x_0)+f^{\prime }(x_0)ih+\frac{f^{″}(\xi )}{2!}(ih{)}^2,}$$and with some algebraic manipulation, this leads to$${\displaystyle \frac{f(x_0+ih)-f(x_0)}{ih}=f^{\prime }(x_0)+\frac{f^{″}(\xi )}{2!}ih,}$$and further noting that the quantity on the left is the approximation from the finite difference method and that the quantity on the right is the exact quantity of interest plus a remainder, clearly that remainder is the local truncation error. A final expression of this example and its order is:$${\displaystyle \frac{f(x_0+ih)-f(x_0)}{ih}=f^{\prime }(x_0)+O(h).}$$

In this case, the local truncation error is proportional to the step sizes. The quality and duration of simulated FDM solution depends on the discretization equation selection and the step sizes (time and space steps). The data quality and simulation duration increase significantly with smaller step size.^[2] Therefore, a reasonable balance between data quality and simulation duration is necessary for practical usage. Large time steps are useful for increasing simulation speed in practice. However, time steps which are too large may create instabilities and affect the data quality.^[3][4]

Thevon Neumann andCourant-Friedrichs-Lewy criteria are often evaluated to determine the numerical model stability.^[3][4][5][6]

For example, consider the ordinary differential equation$${\displaystyle u^{\prime }(x)=3u(x)+2.}$$TheEuler method for solving this equation uses the finite difference quotient$${\displaystyle \frac{u(x+h)-u(x)}{h}\approx u^{\prime }(x)}$$to approximate the differential equation by first substituting it for${\displaystyle u^{\prime }(x)}$ then applying a little algebra (multiplying both sides byh, and then adding${\displaystyle u(x)}$ to both sides) to get$${\displaystyle u(x+h)\approx u(x)+h(3u(x)+2).}$$The last equation is a finite-difference equation, and solving this equation gives an approximate solution to the differential equation.

Consider the normalizedheat equation in one dimension, with homogeneousDirichlet boundary conditions

One way to numerically solve this equation is to approximate all the derivatives by finite differences. First partition the domain in space using a mesh${\displaystyle x_0,\dots ,x_J}$ and in time using a mesh${\displaystyle t_0,\dots ,t_N}$. Assume a uniform partition both in space and in time, so the difference between two consecutive space points will beh and between two consecutive time points will bek. The points

$${\displaystyle u(x_j,t_n)=u_j^n}$$

will represent the numerical approximation of${\displaystyle u(x_j,t_n).}$

Using aforward differenceat time${\displaystyle t_n}$ and a second-ordercentral difference for the space derivative at position${\displaystyle x_j}$ (FTCS) gives the recurrence equation:

$${\displaystyle \frac{u_j^{n+1}-u_j^n}{k}=\frac{u_{j+1}^n-2u_j^n+u_{j-1}^n}{h^2}.}$$

This is anexplicit method for solving the one-dimensionalheat equation.

One can obtain${\displaystyle u_j^{n+1}}$ from the other values this way:

$${\displaystyle u_j^{n+1}=(1-2r)u_j^n+r{u}_{j-1}^n+r{u}_{j+1}^n}$$

where${\displaystyle r=k/h^2.}$

So, with this recurrence relation, and knowing the values at timen, one can obtain the corresponding values at timen+1.${\displaystyle u_0^n}$ and${\displaystyle u_J^n}$ must be replaced by the boundary conditions, in this example they are both 0.

This explicit method is known to benumerically stable andconvergent whenever${\displaystyle r\le 1/2}$.^[7] The numerical errors are proportional to the time step and the square of the space step:$${\displaystyle \mathrm{\Delta }u=O(k)+O(h^2)}$$

Using thebackward differenceat time${\displaystyle t_{n+1}}$ and a second-order central difference for the space derivative at position${\displaystyle x_j}$ (The Backward Time, Centered Space Method "BTCS") gives the recurrence equation:

$${\displaystyle \frac{u_j^{n+1}-u_j^n}{k}=\frac{u_{j+1}^{n+1}-2u_j^{n+1}+u_{j-1}^{n+1}}{h^2}.}$$

This is animplicit method for solving the one-dimensionalheat equation.

One can obtain${\displaystyle u_j^{n+1}}$ from solving a system of linear equations:

$${\displaystyle (1+2r)u_j^{n+1}-r{u}_{j-1}^{n+1}-r{u}_{j+1}^{n+1}=u_j^n}$$

The scheme is alwaysnumerically stable and convergent but usually more numerically intensive than the explicit method as it requires solving a system of numerical equations on each time step. The errors are linear over the time step and quadratic over the space step:$${\displaystyle \mathrm{\Delta }u=O(k)+O(h^2).}$$

Finally, using the central difference at time${\displaystyle t_{n+1/2}}$ and a second-order central difference for the space derivative at position${\displaystyle x_j}$ ("CTCS") gives the recurrence equation:

$${\displaystyle \frac{u_j^{n+1}-u_j^n}{k}=\frac{1}{2}\left(\frac{u_{j+1}^{n+1}-2u_j^{n+1}+u_{j-1}^{n+1}}{h^2}+\frac{u_{j+1}^n-2u_j^n+u_{j-1}^n}{h^2}\right).}$$

This formula is known as theCrank–Nicolson method.

One can obtain${\displaystyle u_j^{n+1}}$ from solving a system of linear equations:

$${\displaystyle (2+2r)u_j^{n+1}-r{u}_{j-1}^{n+1}-r{u}_{j+1}^{n+1}=(2-2r)u_j^n+r{u}_{j-1}^n+r{u}_{j+1}^n}$$

The scheme is alwaysnumerically stable and convergent but usually more numerically intensive as it requires solving a system of numerical equations on each time step. The errors are quadratic over both the time step and the space step:$${\displaystyle \mathrm{\Delta }u=O(k^2)+O(h^2).}$$

To summarize, usually theCrank–Nicolson scheme is the most accurate scheme for small time steps. For larger time steps, the implicit scheme works better since it is less computationally demanding. The explicit scheme is the least accurate and can be unstable, but is also the easiest to implement and the least numerically intensive.

Here is an example. The figures below present the solutions given by the above methods to approximate the heat equation

$${\displaystyle U_t=\alpha U_{xx},\alpha =\frac{1}{{\pi }^2},}$$

with the boundary condition

$${\displaystyle U(0,t)=U(1,t)=0.}$$

The exact solution is

$${\displaystyle U(x,t)=\frac{1}{{\pi }^2}{e}^{-t}\sin (\pi x).}$$

Comparison of Finite Difference Methods

Explicit method (not stable)

Implicit method (stable)

Crank-Nicolson method (stable)

The (continuous)Laplace operator in${\displaystyle n}$-dimensions is given by${\displaystyle \mathrm{\Delta }u(x)=\sum _{i=1}^n{\mathrm{\partial }}_i^{2}u(x)}$.The discrete Laplace operator${\displaystyle {\mathrm{\Delta }}_{h}u}$ depends on the dimension${\displaystyle n}$.

In 1D the Laplace operator is approximated as$${\displaystyle \mathrm{\Delta }u(x)=u^{″}(x)\approx \frac{u(x-h)-2u(x)+u(x+h)}{h^2}=:{\mathrm{\Delta }}_{h}u(x).}$$This approximation is usually expressed via the followingstenciland which represents a symmetric, tridiagonal matrix. For an equidistant grid one gets aToeplitz matrix.

The 2D case shows all the characteristics of the more general n-dimensional case. Each second partial derivative needs to be approximated similar to the 1D casewhich is usually given by the followingstencil

Consistency of the above-mentioned approximation can be shown for highly regular functions, such as${\displaystyle u\in C^4(\mathrm{\Omega })}$.The statement is$${\displaystyle \mathrm{\Delta }u-{\mathrm{\Delta }}_{h}u=\mathcal{O}(h^2).}$$

To prove this, one needs to substituteTaylor Series expansions up to order 3 into the discrete Laplace operator.

Similar tocontinuous subharmonic functions one can definesubharmonic functions for finite-difference approximations${\displaystyle u_h}$$${\displaystyle -{\mathrm{\Delta }}_h{u}_h\le 0.}$$

One can define a generalstencil ofpositive type via

If${\displaystyle u_h}$ is (discrete) subharmonic then the following mean value property holds$${\displaystyle u_h(x_C)\le \frac{\sum _{i\in \{N,E,S,W\}}{\alpha }_i{u}_h(x_i)}{\sum _{i\in \{N,E,S,W\}}{\alpha }_i},}$$where the approximation is evaluated on points of the grid, and the stencil is assumed to be of positive type.

A similarmean value property also holds for the continuous case.

For a (discrete) subharmonic function${\displaystyle u_h}$ the following holds$${\displaystyle \underset{{\mathrm{\Omega }}_h}{max}{u}_h\le \underset{\mathrm{\partial }{\mathrm{\Omega }}_h}{max}{u}_h,}$$where${\displaystyle {\mathrm{\Omega }}_h,\mathrm{\partial }{\mathrm{\Omega }}_h}$ are discretizations of the continuous domain${\displaystyle \mathrm{\Omega }}$, respectively the boundary${\displaystyle \mathrm{\partial }\mathrm{\Omega }}$.

A similarmaximum principle also holds for the continuous case.

The SBP-SAT (summation by parts - simultaneous approximation term) method is a stable and accurate technique for discretizing and imposing boundary conditions of a well-posed linearpartial differential equation using high order finite differences.^[8][9]

The method is based on finite differences where the differentiation operators exhibitsummation-by-parts properties. Typically, these operators consist of differentiation matrices with central difference stencils in the interior with carefully chosen one-sided boundary stencils designed to mimic integration-by-parts in the discrete setting. Using the SAT technique, the boundary conditions of the PDE are imposed weakly, where the boundary values are "pulled" towards the desired conditions rather than exactly fulfilled. If the tuning parameters (inherent to the SAT technique) are chosen properly, the resulting system of ODE's will exhibit similar energy behavior as the continuous PDE, i.e. the system has no non-physical energy growth. This guarantees stability if an integration scheme with a stability region that includes parts of the imaginary axis, such as the fourth orderRunge-Kutta method, is used. This makes the SAT technique an attractive method of imposing boundary conditions for higher order finite difference methods, in contrast to for example the injection method, which typically will not be stable if high order differentiation operators are used.

• K.W. Morton and D.F. Mayers,Numerical Solution of Partial Differential Equations, An Introduction. Cambridge University Press, 2005.
• Autar Kaw and E. Eric Kalu,Numerical Methods with Applications, (2008)[1]. Contains a brief, engineering-oriented introduction to FDM (for ODEs) inChapter 08.07.
• John Strikwerda (2004).Finite Difference Schemes and Partial Differential Equations (2nd ed.). SIAM.ISBN 978-0-89871-639-9.
• Smith, G. D. (1985),Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed., Oxford University Press
• Peter Olver (2013).Introduction to Partial Differential Equations. Springer. Chapter 5: Finite differences.ISBN 978-3-319-02099-0..
• Randall J. LeVeque,Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.
• Sergey Lemeshevsky, Piotr Matus, Dmitriy Poliakov(Eds): "Exact Finite-Difference Schemes", De Gruyter (2016). DOI:https://doi.org/10.1515/9783110491326 .
• Mikhail Shashkov:Conservative Finite-Difference Methods on General Grids, CRC Press, ISBN 0-8493-7375-1 (1996).

• Finite differences
• Numerical differential equations

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