MATH2089 Chap.7 Partial Differential Equations
Partial Differential Equations
Second order PDE classification separates elliptic, parabolic and hyperbolic behaviour through the principal part. The class determines which boundary or initial data are natural and how a finite difference method should be assembled. This chapter derives local stencils, maps them into sparse systems or time updates, handles boundary values and links explicit or implicit schemes to stability and refinement checks.
Topic 7 receives published final exam emphasis. Classification comes before discretisation. For a two-variable second-order principal part, the discriminant built from the second-derivative coefficients distinguishes elliptic, parabolic and hyperbolic behaviour under the course convention. Elliptic problems typically describe equilibrium and are driven by boundary data.
Parabolic problems combine spatial smoothing with time evolution. Hyperbolic problems carry wave-like information through initial data and characteristic directions. The class guides what data are natural and which numerical questions must be asked. A finite-difference stencil replaces a derivative at one grid node by a weighted relationship among neighbours.
The centred Laplacian links a node to axial neighbours and creates a sparse global system because each equation touches only a local set of unknowns. Draw the stencil, label mesh widths and choose a node ordering before writing matrix rows. Boundary neighbours are known values and move to the right-hand side; interior neighbours create matrix coefficients. Time-dependent problems add time levels.
An explicit scheme forms the next state directly from known values and is easy to implement, but its time step may be restricted by mesh width and physical coefficients. For diffusion, a ratio involving the time step divided by the square of the spatial step shows why halving the mesh can require a much smaller time increment.
An implicit scheme places new-time unknowns inside the stencil and requires a sparse solve at each step, often with a reusable matrix factorisation. Stability is not the same as accuracy. A stable method can still be inaccurate on a coarse grid, while a high-order discretisation can fail when its time-step restriction is violated. Refine space and time compatibly and compare at the same physical location and time.
Test boundary values explicitly, because a correct interior relation can be undermined by one indexing or sign error at the edge. Simple exact solutions, symmetry, conservation or known steady states provide valuable verification cases. A complete PDE answer states the class, required data, grid, stencil, assembly rule, solver and stability condition. It then reports a refinement or structural check.
Topic 7 has heavy emphasis in the published Numerical Methods final description, so students should practise both directions: derive a scheme from an equation and infer the equation class, data and stability issue from a printed scheme.
What this chapter covers
- 01
PDE discriminant classification
- 02
Elliptic boundary problems
- 03
Parabolic diffusion problems
- 04
Hyperbolic wave problems
- 05
Five point finite difference stencil
- 06
Sparse assembly and stability
Worked example · free
Assemble a boundary difference system
- +1Set interior unknowns at one third and two thirds.
- +1Apply the centred second difference at the first interior node.
- +1Apply it at the second node and move the known boundary to the right side.
- +1Solve to obtain one third and two thirds, matching the exact linear solution.
Key terms
- Elliptic PDE
- A second order PDE class typically describing spatial equilibrium with boundary data.
- Finite difference stencil
- A local weighted relation among grid values used to approximate derivatives.
- Sparse matrix
- A matrix containing relatively few nonzero entries because interactions are local.
Partial Differential Equations FAQ
What does the PDE discriminant decide?
The sign of B squared minus 4AC classifies the principal part as elliptic, parabolic or hyperbolic. That class signals different data patterns and numerical behaviour.
Why is the global matrix sparse?
Each finite difference equation couples one grid node only to a small local neighbourhood. Every row therefore contains only a few nonzero coefficients even on a large grid.
What must be checked for a time scheme?
Check the time level placement, boundary treatment, consistency and the relevant stability relation between time step, grid spacing and physical parameters. Smooth output alone is not evidence.
Exam move
Practise the sequence classify, state data, draw grid, derive stencil, assemble, solve and verify. Draw the five point stencil from memory and map a small grid to matrix rows.
Give extra time to parabolic and hyperbolic time levels because the official final scope places heavy emphasis on Topic 7. Work one example in both directions: derive a stencil from a PDE, then identify the PDE class and time-level structure from a supplied stencil. Mark boundary nodes in a different colour and show exactly how their known values enter the right-hand side.
Compare an explicit diffusion step with an implicit matrix solve and track what happens to the stability ratio when the spatial mesh is halved. Finish with a manufactured linear or steady solution that the discrete operator should reproduce. This sequence makes classification, indexing, stability and verification visible under exam time pressure.
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