Concept Architecture
Concept
Theoretically, the Finite Difference Method (FDM) is a numerical technique for approximating the solutions of differential equations by replacing continuous derivatives with finite difference approximations evaluated on a discrete computational grid. The method converts ordinary and partial differential equations into systems of algebraic equations that can be solved numerically. It is widely used in scientific computing and provides the mathematical foundation for many simulations involving diffusion, transport and dynamic processes. In health economics, finite difference methods are primarily employed when modelling continuous-time systems governed by differential equations that lack closed-form analytical solutions.
Mathematically, the Finite Difference Method approximates derivatives using values of the dependent variable at neighbouring grid points. Common approximations include forward, backward and central differences, each possessing distinct numerical accuracy and stability properties. Once the differential equation has been discretised, the resulting algebraic equations are solved iteratively or directly to approximate the behaviour of the underlying continuous system.
In practice, finite difference methods are implemented within specialised numerical software and scientific computing environments. In health economics, they are used to solve differential equation models underlying pharmacokinetic simulations, infectious disease transmission models, tumour growth models and physiological systems that provide inputs for decision-analytic and cost-effectiveness models. Appropriate selection of grid spacing and time step is essential to ensure numerical stability, convergence and computational efficiency.
Purpose
Used to obtain numerical solutions to ordinary and partial differential equations by discretising continuous derivatives, enabling simulation of complex dynamic systems that support health economic modelling and health technology assessment.
Mathematical Formulae
Primary Formula
Central difference approximation of the first derivative:
dy/dx � (y??? ? y???) / (2h)
Supporting Formulae
Forward difference:
dy/dx � (y??? ? y?) / h
Backward difference:
dy/dx � (y? ? y???) / h
Second derivative:
d�y/dx� � (y??? ? 2y? + y???) / h�
Related Mathematical Methods
- Runge-Kutta Method
- Euler Method
- Finite Element Method
- Numerical Integration
- Initial Value Problem
- Partial Differential Equation
- Ordinary Differential Equation
Example
A pharmacokinetic diffusion model describes drug concentration throughout tissue using a partial differential equation for diffusion. The tissue is divided into equally spaced spatial intervals of 0.5 mm, and second derivatives are approximated using central finite differences. Solving the resulting algebraic equations produces an approximation of drug concentration at each location, allowing estimation of treatment effectiveness and downstream cost-effectiveness outcomes.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| INDEX | =(INDEX(B:B,A2+1)-INDEX(B:B,A2-1))/(2*$F$1) | Calculates a central finite difference approximation. |
| LET | =LET(h,$F$1,(C2-A2)/(2*h)) | Improves readability of finite difference calculations. |
| SEQUENCE | =SEQUENCE(201,1,0,0.5) | Generates evenly spaced computational grid points. |
| ABS | =ABS(B2-C2) | Calculates numerical approximation error. |
| MMULT | Matrix solution of discretised equations | Solves systems of algebraic equations arising from finite difference discretisation. |
VBA (Optional)
VBA can automate finite difference simulations by iteratively solving discretised differential equations across large spatial and temporal grids.
Sources
- LeVeque RJ. Finite Difference Methods for Ordinary and Partial Differential Equations. Society for Industrial and Applied Mathematics.
- Smith GD. Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford University Press.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- Burden RL, Faires JD. Numerical Analysis. Cengage Learning.
Related Concepts (5)
Library
Publications
1
Numerical Analysis of Parameterized Nonlinear Equations — Werner C. Rheinboldt, 1st Edition ed., 1986 (John Wiley & Sons)
A specialist numerical-analysis reference on iterative solution, continuation and computation for parameterized nonlinear systems, linked here to the ontology's numerical-method and optimization branch.
BookView source →
Frequently Asked Questions (6)
What is the finite difference method?
A numerical method that approximates derivatives using finite differences to solve differential equations.
Source: LeVeque RJ. Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM; 2007. doi:10.1137/1.9780898717839.
How does the finite difference method solve differential equations?
The finite difference method solves differential equations by approximating the derivatives in them using finite differences, that is, differences of function values at nearby points. Replacing each derivative with such an approximation turns the differential equation into a set of algebraic relations that can be solved numerically. This substitution of finite differences for derivatives is how the finite difference method proceeds By swapping each derivative for a difference of nearby values, the method turns a differential equation into algebraic relations a computer can solve at a set of discrete points.
Source: LeVeque RJ. Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM; 2007. doi:10.1137/1.9780898717839.
What does the finite difference method use in place of derivatives?
The finite difference method uses finite differences in place of derivatives, approximating each derivative by a difference of function values at neighbouring points. Because exact derivatives cannot be handled directly by a numerical scheme, they are replaced with these discrete approximations. Using finite differences instead of derivatives is the defining step of the finite difference method By swapping each derivative for a difference of nearby values, the method turns a differential equation into algebraic relations a computer can solve at a set of discrete points.
Source: LeVeque RJ. Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM; 2007. doi:10.1137/1.9780898717839.
Why does the finite difference method approximate derivatives?
The finite difference method approximates derivatives because a differential equation cannot be solved numerically while it contains exact derivatives, so those derivatives are replaced with finite differences of function values. This turns the problem into algebraic equations that a computer can solve. Approximating the derivatives is therefore the means by which the finite difference method makes a differential equation tractable By swapping each derivative for a difference of nearby values, the method turns a differential equation into algebraic relations a computer can solve at a set of discrete points.
Source: LeVeque RJ. Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM; 2007. doi:10.1137/1.9780898717839.
What does the finite difference method produce from a differential equation?
The finite difference method produces a set of algebraic relations from a differential equation, obtained by approximating its derivatives using finite differences. Solving these relations gives approximate values of the solution at discrete points. This conversion of a differential equation into solvable discrete equations is the outcome of applying the finite difference method By swapping each derivative for a difference of nearby values, the method turns a differential equation into algebraic relations a computer can solve at a set of discrete points.
Source: LeVeque RJ. Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM; 2007. doi:10.1137/1.9780898717839.
How does the finite difference method relate to the central difference?
The finite difference method approximates derivatives using finite differences to solve differential equations, while the central difference is a finite-difference method that approximates a derivative using function values on either side of the point of interest. The central difference is one particular finite difference the method can use. The two are connected, since the central difference supplies one of the derivative approximations on which the finite difference method draws By swapping each derivative for a difference of nearby values, the method turns a differential equation into algebraic relations a computer can solve at a set of discrete points.
Source: LeVeque RJ. Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM; 2007. doi:10.1137/1.9780898717839.
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 2 Apr 2026
Content version: 1.0.0
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