"In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences."
Numerical methods used to discretize PDEs and solve them on a computer.
Difference operators: A difference operator is a mathematical operator that maps a sequence of numbers into another sequence of numbers. It is used to represent finite difference approximations of derivatives.
Taylor series: A Taylor series is a representation of a function as an infinite sum of terms that are each multiplied by the values of the function's derivatives evaluated at a single point.
Approximation of derivatives: The approximation of derivatives involves using finite difference methods to estimate the values of derivatives of a function at specific points.
Boundary value problems: A boundary value problem is a differential equation that is subject to conditions at two different points. It requires the solution of the differential equation subject to these conditions.
Initial value problems: An initial value problem is a differential equation that is subject to an initial condition. It requires the solution of the differential equation subject to this initial condition.
Explicit and implicit methods: Finite difference methods can be classified into two main types: Explicit and implicit methods. An explicit method computes the solution at each time step using the solution from the previous time step, while an implicit method includes the unknown solution in the equation to be solved.
Consistency, stability, and convergence: These three properties are important for the accuracy and reliability of finite difference methods. Consistency refers to the degree to which the finite difference approximation approaches the exact solution, stability refers to the tendency of the solution to remain bounded over time, and convergence is the convergence of the finite difference solution to the exact solution as the grid size approaches zero.
Alternating direction implicit method: The alternating direction implicit method is a technique for solving partial differential equations in two or more dimensions. It involves breaking the problem into smaller, simpler subproblems and solving them iteratively.
Finite element method: The finite element method is a numerical technique for solving differential equations. It involves discretizing the differential equation into smaller subdomains, approximating the unknown solution with a piecewise polynomial function, and solving for the unknown coefficients.
Multigrid methods: Multigrid methods are iterative methods for solving linear and nonlinear systems of equations. They involve solving the problem on a sequence of grids with different resolutions, and using the solution on coarse grids to improve the solution on finer grids.
Forward Finite Difference Method (FDM): Approximates the derivative of a function at discrete points using the forward difference formula where the derivative at a point is calculated using the values of the function at that point and the next point.
Central Finite Difference Method (CFDM): Approximates the derivative using a central point and two neighboring points in both directions.
Backward Finite Difference Method (BFDM): Approximates the derivative using the backward difference formula, which involves the value of the function at a point and the previous point.
Implicit Finite Difference Method: The implicit finite difference method is used when the change is not small, or the function is non-linear. This method involves solving the system of equations derived from the difference formulae.
Explicit Finite Difference Method: The explicit finite difference method is used when the change is small, and the function is linear. The derivative at each point is calculated using the values at that point and its neighbors.
Crank-Nicolson Method: Crank-Nicolson is a numerical method that combines both the implicit and explicit methods, making it more accurate than either method on its own. It involves solving a system of linear equations.
Adams-Bashforth Method: The Adams-Bashforth method is a linear multistep method that involves approximating future values of a function based on past values. It requires the initial values and the first few derivatives of the function to calculate the future values.
Adams-Moulton Method: The Adams-Moulton method is similar to the Adams-Bashforth method but involves using the values of the function at the new time step as well.
Runge-Kutta Method: The Runge-Kutta method or RK method is a family of methods used to solve ordinary differential equations. It involves approximating the solution at intermediate points using weighted averages.
Finite Volume Method: The finite volume method is a method used to solve partial differential equations. It involves dividing the domain of the equation into small control volumes and developing the equations at the boundaries of these control volumes.
Finite Element Method: The finite element method is a method used to solve partial differential equations. It involves dividing the domain of the equation into small elements and approximating the solution within each element using piecewise functions.
"Approximating derivatives with finite differences."
"Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps."
"The spatial domain...broken into a finite number of steps."
"The value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points."
"Finite difference methods convert ordinary differential equations (ODE) or partial differential equations (PDE), which may be nonlinear, into a system of linear equations."
"Finite difference methods convert...into a system of linear equations."
"...that can be solved by matrix algebra techniques."
"Modern computers can perform these linear algebra computations efficiently..."
"...which, along with their relative ease of implementation, has led to the widespread use of FDM in modern numerical analysis."
"Today, FDM are one of the most common approaches to the numerical solution of PDE, along with finite element methods."
"...numerical solution of PDE."
"Finite-difference methods convert ODE or PDE into a system of linear equations, while finite element methods approach the problem by discretizing the domain into a finite number of elements."
"Finite difference methods...can be solved by matrix algebra techniques."
"The value of the solution at these discrete points is approximated..."
"Finite difference methods convert...which may be nonlinear, into a system of linear equations."
"Differential equations by approximating derivatives with finite differences."
"Both the spatial domain and time interval (if applicable) are discretized..."
"The value of the solution at...points is approximated by solving algebraic equations containing finite differences..."
"...relative ease of implementation, has led to the widespread use of FDM in modern numerical analysis."