sofia.pde.jacobian

class sofia.pde.jacobian.Jacobian(*args)

Bases: Function

The symbolic Jacobian of a system of equations.

Parameters:

  • exprs – The residual expressions of the system, or a single expression.

  • variables – The fields to differentiate exprs with respect to, or a single field.

Raises:

  • TypeError – If variables isn’t one or more GridField.

  • ValueError – If the variables does not contain unique fields.

property entries: dict[tuple[int, int, tuple[int, ...]], Expr]

Compute the Jacobian entries of the system of equations.

is_nearest_neighbor(axis: int) → bool

Whether entries only couples neighbors along axis.

to_block_tridiagonal(axis: int) → tuple[MutableDenseMatrix, MutableDenseMatrix, MutableDenseMatrix, tuple[Expr, ...]]

Return blocks of the Jacobian along axis as (D, A, C, b).

D is the diagonal, A is the lower diagonal, and C is the upper diagonal.

to_block_tridiagonal_solvable(axis: int, outputs: tuple[GridField, ...] | None = None, negate: bool = True) → tuple[tuple[GridField, ...], tuple[MutableDenseMatrix, MutableDenseMatrix, MutableDenseMatrix], tuple[Expr, ...]]

Return (x, exprs, b) as expected by ir.SolveBlockTridiagonal.

Example usage: ir.SolveBlockTridiagonal(*Jacobian(exprs, variables).to_block_tridiagonal_solvable(axis)).

Parameters:

  • axis – The grid axis to build block-tridiagonal system along.

  • outputs – Fields to hold the Newton step for each field in variables, in the same order.

  • negate – Controls the sign of b. By default b = -fixed_exprs, matching the Newton step J @ dx = -R.