---
title: "Finite difference method; explicit and implicit schemes"
subject: "Computational Techniques in Civil Engineering"
chapter: 5
key: "semester_8/computational_techniques_in_civil_engineering:fdm-explicit-implicit"
source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/fdm-explicit-implicit
site: BCT Engineering
---

# Finite difference method; explicit and implicit schemes

Computational Techniques in Civil Engineering · Chapter 5: Finite difference method

- Asked in: 2077 Chaitra, 2+3 marks; 2078 Chaitra, 2+1+1 marks (asked 2 times)
- Video: [6. Implicit Method Concept and Notation - PDE Solutions || Safayat Munna,BUET'19](https://www.youtube.com/watch?v=J9YAniLyq0Y) — Safayat Munna
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/fdm-explicit-implicit
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-5

## Question

What is the finite difference method? Explain explicit and implicit finite difference schemes with examples.

## Short answer

The finite difference method replaces derivatives with differences. In explicit schemes, spatial terms use known time level, giving direct solution but requiring C ≤ 1 for stability. In implicit schemes, spatial terms use unknown time level, requiring matrix solution but remaining unconditionally stable.

## Answer

The **finite difference method (FDM)** is a numerical technique that replaces the continuous solution domain by a grid of discrete points and replaces the derivatives in a differential equation by differences between the values at neighbouring grid points. The differential equation becomes a set of algebraic equations in the unknown grid values.

#### Explicit and implicit schemes (example: $\partial Q/\partial t+c\,\partial Q/\partial x=0$, $C=c\Delta t/\Delta x$)
**Explicit:** the spatial derivative is evaluated at the *known* time level $n$.
$$
\frac{Q_i^{n+1}-Q_i^n}{\Delta t}+c\,\frac{Q_i^n-Q_{i-1}^n}{\Delta x}=0\ \Rightarrow\ Q_i^{n+1}=Q_i^n-C\,(Q_i^n-Q_{i-1}^n)
$$
Each unknown is found directly from known values. It is conditionally stable (needs $C\le1$).

**Implicit:** the spatial derivative is evaluated at the *new* time level $n+1$.
$$
\frac{Q_i^{n+1}-Q_i^n}{\Delta t}+c\,\frac{Q_i^{n+1}-Q_{i-1}^{n+1}}{\Delta x}=0\ \Rightarrow\ Q_i^{n+1}=\frac{Q_i^n+C\,Q_{i-1}^{n+1}}{1+C}
$$
The unknown at point $i$ depends on other unknowns at $n+1$. For problems such as groundwater flow the result is a set of simultaneous equations (tridiagonal matrix). It is unconditionally stable.

#### Comparison

| Point | Explicit | Implicit |
|---|---|---|
| Spatial terms at | known level $n$ | unknown level $n+1$ |
| Solution | direct, point by point | simultaneous equations (matrix) |
| Effort per step | small | larger |
| Stability | conditional ($C\le1$) | unconditional (large $\Delta t$ allowed) |
| Typical use | dynamic-wave models with small $\Delta t$ | groundwater, kinematic wave, long simulations |

**Example (groundwater, $\lambda=T\Delta t/S\Delta x^2$):** explicit $h_i^{n+1}=h_i^n+\lambda(h_{i+1}^n-2h_i^n+h_{i-1}^n)$ needs $\lambda\le\tfrac12$; implicit $-\lambda h_{i-1}^{n+1}+(1+2\lambda)h_i^{n+1}-\lambda h_{i+1}^{n+1}=h_i^n$ is stable for any $\lambda$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does the finite difference method do to a differential equation?
   A. Fits one polynomial over the whole domain
   B. Solves it analytically on a grid
   C. Replaces derivatives by differences of grid-point values, giving algebraic equations
   D. Removes the time derivative
   Answer: C. FDM covers the domain with a grid and approximates derivatives with differences, converting the PDE into algebraic equations. (Key point: Idea of FDM)

2. In an implicit scheme, at which time level is the spatial derivative evaluated?
   A. At the known level n
   B. At the new, unknown level n+1
   C. Halfway between n and n-1
   D. At the initial time only
   Answer: B. Implicit schemes use level n+1, so unknowns are coupled and need simultaneous equations. (Key point: Explicit vs implicit)

3. Which statement on stability is correct for the advection example?
   A. Implicit is conditionally stable, explicit is unconditional
   B. Both are unconditionally stable
   C. Explicit is conditionally stable (C <= 1), implicit is unconditionally stable
   D. Neither needs a stability condition
   Answer: C. The explicit scheme needs Courant number C <= 1, while the implicit one allows large delta t. (Key point: Stability)

4. What limit applies to the explicit groundwater scheme h_i^{n+1} = h_i^n + lambda(h_{i+1} - 2h_i + h_{i-1})?
   A. lambda >= 1
   B. lambda <= 1/2
   C. lambda = 2
   D. No limit
   Answer: B. The explicit scheme is stable only for lambda = T dt/(S dx^2) <= 1/2, whereas the implicit one works for any lambda. (Key point: Groundwater example)

## Related in Computational Techniques in Civil Engineering

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