10: Variable Bathymetry Support

[philipnickel]

Description

Extend the solver to support spatially varying water depth h(x). This is required for the submerged bar benchmark.

Background

With variable bathymetry:

• d(x,t) = η(x,t) + h(x)
• K matrix components now vary with x through both η and h
• May want to align element boundaries with bathymetry discontinuities (slope changes)

Tasks

• Modify depth from scalar to function/array h(x)
• Update d(x) computation to use h(x)
• Update K matrix computation
• Implement piecewise linear bathymetry definition
• Test with simple sloping bottom
• Test with bar geometry

Acceptance Criteria

• Constant h(x) = h₀ gives same result as before
• Sloping bottom: solution remains physical (no instability)
• Bar geometry: waves shoal and steepen as expected
• No spurious reflections from bathymetry changes

Technical Notes

• Consider placing element boundaries at slope transitions
• The bottom BC ∂ₙΦ = 0 should still be natural in weak form
• Sharp bathymetry changes may need special treatment