Lesson 1: Spatial and Temporal Grids

1 Modelling context

A partial differential equation describes how a quantity varies in space and time. The heat equation, the wave equation, and the diffusion equation are all stated on a continuous domain, with the solution defined at every point. To compute a numerical solution, the continuous domain must be replaced by a finite set of points.

The choice of grid is not a technical detail. It defines what the numerical problem actually is, and it determines the accuracy, the stability, and the cost of the solution. Two grids of different resolution produce two different discrete problems, even when the underlying PDE is the same.

This lesson establishes the grid conventions used throughout Module 3, and the documentation discipline that makes a grid choice reviewable and reproducible.

The grid is part of the model

Every simulation has a grid, and every grid is a decision. The grid spacing must be chosen deliberately, justified in terms of the problem and the accuracy required, and documented alongside the parameters and boundary conditions.

2 Learning objectives

By the end of this lesson you will be able to

Construct a uniform spatial grid and a uniform temporal grid for a PDE problem.
Distinguish grid points from the midpoints between them.
Document the grid so that a reader can reconstruct it exactly.
Explain how grid resolution affects the accuracy of a numerical solution.
Estimate the computational cost of a grid refinement before running it.

3 Prerequisite review

Domain: The region of space and time over which the PDE holds. For a one-dimensional problem, typically an interval [a, b] in space and [0, T] in time.
Grid spacing: The distance between consecutive grid points. Denoted Δx in space and Δt in time.
Uniform grid: A grid in which the spacing is constant. Uniform grids simplify the finite-difference formulas and are the default in this course.
Grid function: The discrete representation of the solution, denoted u_i^n for the value at spatial index i and temporal index n.

4 Mathematical formulation

A uniform one-dimensional grid

Suppose the spatial domain is the interval [a, b]. Divide it into N_x equal subintervals of width

Δx = (b − a) / N_x

The grid points are

x_i = a + i Δx,    i = 0, 1, …, N_x

Similarly, for the time interval [0, T] divided into N_t equal steps of size Δt = T / N_t, the time levels are

t_n = n Δt,    n = 0, 1, …, N_t

The grid function

The numerical solution is a grid function u_i^n, defined at every point (x_i, t_n). It approximates the exact solution:

u_i^n ≈ u(x_i, t_n)

The finite-difference method provides a rule for computing u_i^{n+1} from values at level n — a time-marching scheme. The initial condition supplies u_i^0 for all i; the boundary conditions supply values or relations at i = 0 and i = N_x for all n.

Grid resolution and accuracy

The accuracy of the numerical solution depends on both grid spacings. For a scheme with temporal truncation error O(Δt^p) and spatial truncation error O(Δx^q), the global error scales approximately as

E ≈ C_t Δt^p + C_x Δx^q

where C_t and C_x are constants that depend on the problem and the scheme. Both terms must be controlled to achieve a target accuracy. Reducing one without the other will eventually produce diminishing returns.

Computational cost

For a one-dimensional problem, the total number of grid points is approximately

N_x · N_t = ( (b − a) / Δx ) · ( T / Δt )

The cost of a simulation is proportional to this product, multiplied by the cost per grid point (which depends on the scheme). For explicit schemes, the CFL stability restriction imposes an additional relation between Δt and Δx, further increasing the cost of grid refinement.

Figure 3.3. A finite-difference grid. The bottom row is supplied by the initial condition; the left and right columns by the boundary conditions. The interior points are computed by the scheme, one time level at a time.

5 Worked example

Laying out a grid for the heat equation

Consider the heat equation on the spatial interval [0, 1] and the time interval [0, 0.5]:

∂u/∂t = α ∂²u/∂x²,    α = 0.1

Initial condition: u(x, 0) = sin(π x). Boundary conditions: u(0, t) = 0, u(1, t) = 0.

Step 1 — Choose the spatial grid.

Let N_x = 20. Then

Δx = 1 / 20 = 0.05
x_i = i · 0.05,    i = 0, 1, …, 20

Step 2 — Choose the time step.

The CFL condition for the explicit scheme is Δt ≤ Δx² / (2α). Substituting:

Δt ≤ (0.05)² / (2 · 0.1) = 0.0025 / 0.2 = 0.0125

Choose Δt = 0.01, which is comfortably below the limit. Then

N_t = 0.5 / 0.01 = 50
t_n = n · 0.01,    n = 0, 1, …, 50

Step 3 — Determine the grid dimensions.

The grid function has dimensions (N_x + 1) × (N_t + 1) = 21 × 51, giving 1071 grid points. The unknown values are the interior points at time levels 1 through 50: 19 · 50 = 950 values to be computed.

Step 4 — Document the grid.

The documentation for this grid includes:

Spatial domain: [0, 1].
Number of spatial intervals: N_x = 20.
Spatial spacing: Δx = 0.05.
Temporal domain: [0, 0.5].
Time step: Δt = 0.01.
Number of time steps: N_t = 50.
Diffusion number: r = α Δt / Δx² = 0.4.
Stability margin: r = 0.4 < 0.5, stable.

6 Verification check

Checks to perform

Confirm grid dimensions. The number of grid points in space is N_x + 1 (inclusive endpoints), not N_x. An off-by-one error here is the most common source of index errors in PDE code.
Check that the last grid point coincides with the domain endpoint. Confirm x_{N_x} = b to within floating-point tolerance.
Verify the time levels reach the final time exactly. Confirm t_{N_t} = T, or document any endpoint mismatch.
Check the CFL condition before running. Compute r = α Δt / Δx² and confirm it is below the stability limit for the scheme.
Test the grid with a known solution. For u(x, 0) = sin(π x), the exact solution is u(x, t) = e^{−α π² t} sin(π x). Compare the numerical and exact solutions at a few grid points.

7 Interpretation

The grid is the discrete analogue of the continuum. Every quantity that is a function of x or t in the continuous problem becomes an indexed array in the discrete problem. Choosing the grid therefore determines the shape of every data structure in the implementation.

The resolution of the grid determines what features of the solution can be represented. A feature of spatial extent L requires several grid points to be resolved adequately: at minimum, Δx ≪ L. A boundary layer, a sharp front, or an oscillation with wavelength λ requires even more points: Δx ≲ λ / 10 for reasonable accuracy.

Refinement is not free

Refining the grid improves accuracy but increases cost. For an explicit scheme, the CFL condition ties the time step to the spatial step, so refining Δx by a factor of 2 increases the total number of grid operations by a factor of 8. Planning the resolution before running is essential for problems that would otherwise exceed the available computational budget.

8 Uncertainty and limitations

The grid introduces two distinct sources of error into the numerical solution.

Errors introduced by the grid
Source	Cause	Reduced by
Spatial truncation error	Finite-difference approximation of spatial derivatives	Smaller `Δx`
Temporal truncation error	Finite-difference approximation of the time derivative	Smaller `Δt`
Representation error	Features smaller than the grid spacing cannot be represented	Smaller `Δx` until the feature is resolved

The third source is often overlooked. Even a formally accurate scheme cannot represent a solution feature that is smaller than the grid spacing. If the initial condition contains a sharp front and the grid is coarse, the front will be smeared across several grid cells regardless of the scheme's theoretical accuracy.

Resolution matters as much as order

A high-order scheme on a grid that does not resolve the solution is no better than a low-order scheme on the same grid. Before refining the order, ensure the grid resolves the features of interest.

9 Common implementation errors

Watch for these

Off-by-one in the number of grid points. A spatial domain divided into N_x intervals has N_x + 1 grid points, including both endpoints. Forgetting the endpoints produces either a short array or an index error.
Assuming the last grid point reaches the endpoint exactly. Rounding in Δx can produce x_{N_x} = b ± ε. Document any mismatch, or set the last grid point explicitly.
Mixing Δx and Δx² in the implementation. The second difference divides by Δx², the first difference by Δx. A common error is to divide both by Δx.
Forgetting that interior points are the only unknowns. Boundary values are supplied by the boundary conditions at each time level. They are not computed by the scheme.
Choosing a grid that violates the CFL condition. Verify the stability condition before running, not after the solution has blown up.
Documenting only N_x and not Δx. Both are needed for a reader to reconstruct the grid. The same applies to time.

10 Independent modelling task

Your turn

Choose a one-dimensional PDE problem. A simple heat-equation problem on [0, 1] is sufficient.
Choose a spatial resolution N_x. State the domain and justify the resolution in terms of the features of the initial condition you want to represent.
Compute the CFL limit for the explicit scheme. Choose a time step Δt that is below this limit, and compute the resulting diffusion number r.
Compute the grid dimensions (N_x + 1) × (N_t + 1) and the number of unknown values.
Document the grid: domain, N_x, Δx, N_t, Δt, r, stability status.
Refine the grid by a factor of 2 in Δx. Compute the new CFL limit and the new grid dimensions. Compute the ratio of total grid operations to the previous configuration.

This exercise forms part of Laboratory 5 and the grid-documentation component of Assignment 3.

11 Reflection prompt

Reflect before continuing

Halving Δx requires quartering Δt to maintain stability. Does this mean that halving the spatial grid spacing is always a bad idea?

Two or three sentences. Consider the circumstances under which the accuracy gain justifies the cost, and the circumstances under which it does not.

12 Lesson summary

Key takeaways

A grid is a modelling decision. It defines the discrete problem and determines the accuracy, stability, and cost of the solution.
A domain of N_x intervals has N_x + 1 grid points. Both endpoints are included. The interior points are the unknowns.
The CFL condition ties Δt to Δx². Refining the spatial grid by a factor of 2 requires quartering the time step, increasing total operations by a factor of 8.
Grid resolution determines what features can be represented. A feature smaller than the grid spacing cannot be captured, regardless of the scheme's theoretical order.
Document the grid completely. Domain, spacing, number of intervals, diffusion number, and stability status are all part of the numerical problem.

Coming next

Lesson 2 derives the finite-difference approximations that define the stencils applied on the grid constructed here.

§ Self-quiz, practice, and homework

Practice exercises

For a domain [0, 2] divided into N_x = 40 intervals, compute Δx, list the grid points, and count them.
For the same domain, choose r = 0.4 and α = 0.05. Compute Δt and N_t for a final time T = 1.
Halve Δx and recompute Δt, N_t, and the total number of grid operations. Compare.
Sketch a grid with N_x = 5 and N_t = 3. Mark which points are given by the initial condition, by the boundary conditions, and by the scheme.

Homework problems

Grid design report. Choose a one-dimensional PDE problem. Design a grid appropriate to the problem: state the domain, N_x, Δx, the CFL limit, Δt, N_t, and r. Justify each choice in one sentence. Submit a two-page report.
Refinement cost analysis. For the grid above, compute the total cost at three successive refinements (Δx, Δx/2, Δx/4). Plot the cost against Δx on a log-log scale. Estimate the exponent.
Resolution test. For a problem with a sharp initial feature (e.g. a narrow Gaussian or a step function), demonstrate numerically that the feature cannot be resolved on a coarse grid, and show how refinement recovers it.

Mini project

Design a grid for a real PDE problem of your choice — heat conduction in a rod, diffusion of a pollutant, or similar. Document the domain, the grid resolution, the CFL limit, the chosen Δt, and the stability margin. Run the same problem at three resolutions and compare the results. Submit a report and the spreadsheet. This is the basis for Laboratory 5 and Assignment 3.