Finite difference method; explicit and implicit schemes

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

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 $\lambda$.