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

# FDM basic steps; explicit and implicit schemes

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

- Asked in: 2075 Bhadra, 2+2+2 marks
- Video: [Implicit vs Explicit Scheme](https://www.youtube.com/watch?v=1HM8JkkiVIQ) — Knowledge Engineering
- 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-steps-explicit-implicit
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-5

## Question

Describe the basic steps in the finite difference method. Explain explicit and implicit schemes in the finite difference method using suitable examples and expressions.

## Short answer

Basic steps: state PDE/IC/BC, discretize grid, approximate derivatives with finite differences, form algebraic equations, impose initial/boundary values, solve sequentially (explicit) or as simultaneous equations (implicit). Explicit example: Q_i^{n+1}=Q_i^n-C(Q_i^n-Q_{i-1}^n), stable C≤1. Implicit: Q_i^{n+1}=(Q_i^n+C Q_{i-1}^{n+1})/(1+C), unconditionally stable.

## Answer

#### Basic steps in the finite difference method
1. **State the problem:** write the governing PDE with its domain, initial conditions (at $t=0$) and boundary conditions.
2. **Discretise the domain:** draw a grid with spacing $\Delta x$ and time step $\Delta t$; points are $(x_i,t^n)$.
3. **Approximate the derivatives:** replace each derivative by a forward, backward or central difference obtained from the Taylor series.
4. **Write the algebraic equation** at every grid point (explicit or implicit form).
5. **Insert the initial and boundary values** into the equations.
6. **Solve:** march forward in time (explicit), or solve the simultaneous equations at each time step (implicit, e.g. by the Thomas algorithm).
7. **Check** consistency, accuracy (truncation error), stability (Courant number) and convergence by refining $\Delta x,\Delta t$.

```text
 t
 n+1 : o   o   o   o      unknown
 n   : o   o   o   o      known (initial / previous step)
 n-1 : .   .   .   .
       i-1 i  i+1 i+2    --> x
```

#### 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.

**Remark:** with the implicit form the new value at $i$ also depends on the unknown $Q_{i-1}^{n+1}$; for the one-way kinematic wave this is solved by sweeping from the upstream boundary, while for diffusion-type equations a tridiagonal system must be solved.

## Check yourself

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

1. In an explicit scheme the spatial derivative is evaluated at:
   A. the new time level n+1
   B. the known time level n
   C. the average of n-1 and n+1
   D. a boundary point only
   Answer: B. Using only known values at level n lets each unknown be found directly. (Key point: Explicit scheme)

2. What characterises an implicit scheme?
   A. Each unknown is computed directly from known values
   B. The unknown depends on other unknowns at n+1, giving simultaneous equations
   C. It needs Courant number C <= 1
   D. It ignores initial conditions
   Answer: B. The spatial difference is taken at n+1, so values at the new level are coupled (e.g. tridiagonal system). (Key point: Implicit scheme)

3. Compared with the explicit upwind example, the implicit example is:
   A. conditionally stable, needing C <= 1
   B. unconditionally stable
   C. always unstable
   D. stable only for C > 1
   Answer: B. The explicit form needs C <= 1 while the implicit one is unconditionally stable. (Key point: Stability)

4. In the finite difference method, derivatives are replaced by differences obtained from:
   A. Fourier series
   B. Taylor series
   C. Laplace transform
   D. Newton's law of cooling
   Answer: B. Forward, backward or central differences come from Taylor series expansion. (Key point: Derivative approximation)

## Related in Computational Techniques in Civil Engineering

- [Finite difference method; explicit and implicit schemes](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Afdm-explicit-implicit) — Computational Techniques in Civil Engineering, chapter 5, asked 2×
- [First and second order finite difference schemes](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Afd-first-second-order-schemes) — Computational Techniques in Civil Engineering, chapter 5, asked 2×
- [Basic steps in finite element analysis](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Afem-basic-steps) — Computational Techniques in Civil Engineering, chapter 1

## Related in other subjects

- [Explicit, implicit and Crank-Nicolson schemes](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Aexplicit-implicit-crank-nicolson) — Applied Computational Fluid Dynamics, chapter 3