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Neumann (gradient) boundary conditions

Supports: 1D and 2D (selected automatically based on domain)

1D

Specifies the normal gradient at the two boundaries:

where \( g_L \) is the gradient at the left (outward-normal) and \( g_R \) at the right.

2D

The same two parameters apply to all four edges:

• Left edge (x = 0): \( f_{i,0} = f_{i,1} - g_L \cdot \Delta x \)
• Right edge (x = L): \( f_{i,-1} = f_{i,-2} + g_R \cdot \Delta x \)
• Bottom edge (y = 0): \( f_{0,j} = f_{1,j} - g_L \cdot \Delta y \)
• Top edge (y = L): \( f_{-1,j} = f_{-2,j} + g_R \cdot \Delta y \)

gradient_left is used for the “min” boundaries (left, bottom); gradient_right for the “max” boundaries (right, top).

YAML

boundary:
  type: neumann
  gradient_left: 0.0
  gradient_right: 0.0