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 domain and time domain (if applicable) are discretized, 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.
Modern computers can perform these linear 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 with finite element methods.
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 the local truncation error of a method.
The remainder term of the Taylor polynomial can be used to analyze local truncation error.
In this case, the local truncation error is proportional to the step sizes.
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] The von Neumann and Courant-Friedrichs-Lewy criteria are often evaluated to determine the numerical model stability.
The Euler method for solving this equation uses the finite difference quotient
Consider the normalized heat equation in one dimension, with homogeneous Dirichlet boundary conditions
One way to numerically solve this equation is to approximate all the derivatives by finite differences.
This is an explicit method for solving the one-dimensional heat equation.
(The Backward Time, Centered Space Method "BTCS") gives the recurrence equation:
This is an implicit method for solving the one-dimensional heat equation.
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.
The figures below present the solutions given by the above methods to approximate the heat equation
Consistency of the above-mentioned approximation can be shown for highly regular functions, such as
To prove this, one needs to substitute Taylor Series expansions up to order 3 into the discrete Laplace operator.
where the approximation is evaluated on points of the grid, and the stencil is assumed to be of positive type.
A similar mean value property 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 partial differential equation using high order finite differences.
[8][9] The method is based on finite differences where the differentiation operators exhibit summation-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.
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.
