Finite Differences

Approximates derivatives and field equations using discrete values on a grid.

Conceptual context

Many physical fields, including temperature, electric potential, and string displacement, are described as continuous functions. A computer, however, cannot manipulate infinitely many points. It works with discrete grid nodes. Finite differences bridge these descriptions by replacing derivatives with combinations of $u_{i-1}$, $u_i$, and $u_{i+1}$, separated by a $h$. The key question is not merely whether the calculation returns a number, but how closely that number represents the continuous operator. The discrepancy is the $E_T$, and its behavior under grid refinement reveals the actual quality of the method.

🟢 Essential level

A finite difference estimates how a field changes by comparing nearby grid samples. If the value to the right exceeds the central value, the estimated $u'$ is usually positive; if it is smaller, the estimate is usually negative.

Curvature is identified by comparing the central value with both neighbors, which gives an estimate of $u''$. The $h$ controls detail: a fine grid resolves small variations, whereas a coarse grid smooths or misses them. Reducing the step usually decreases $E_T$, but the rate depends on the method order. A higher-order scheme uses refinement more efficiently, although it may require more nodes, careful boundary treatment, and extra computation. The goal is to find a region where the numerical result converges predictably.

Choosing a one-sided, centered, or higher-order scheme is not only a precision question: it also determines how many neighbors are required and how nodes close to a boundary are treated. A centered scheme is usually preferred when both neighbors are available, while near an edge a one-sided formula may be unavoidable even though its $E_T$ is larger.

🔵 Formal level

Basic formulas

The one-sided estimate of $u'$ uses $u_i$, $u_{i+1}$, and $h$:

First-order forward difference

The centered estimate uses $u_{i-1}$ and $u_{i+1}$ on opposite sides of the node. Symmetry cancels lower-order Taylor terms and improves accuracy:

Second-order centered difference

The three-point stencil for $u''$ compares the central value with its neighbors:

Centered second derivative

The asymptotic behavior of $E_T$ is summarized by a scaling law involving $C$, $h$, and $p$:

Truncation-error scaling

Real student questions

Is a finer grid always better? No. It reduces $E_T$ for a consistent scheme, but increases cost and may expose roundoff or instability.

Why is a centered formula usually more accurate? It uses symmetric information and cancels selected terms in the local expansion.

Is $E_T$ the simulation error? It is only one contribution. Modeling, boundary, iterative, data, and roundoff errors also matter.

How do I know whether the expected order is reached? Build a log-log graph of error versus $h$ and look for a stable slope.

Why does $CFL$ appear in a spatial topic? In evolutionary equations, spatial refinement changes the relation between physical transport and time advancement.

Final synthesis

Finite differences turn continuous operators into nodal relations. Their reliability depends on smoothness, grid design, boundaries, and stability. The $E_T$ decreases with $h$ according to $p$ only in the appropriate regime. Good refinement means verifying convergence, not merely multiplying nodes.