The current implementation of the MOZART LU decomposition scheme accepts separate A, L, and U matrices. Add an LU decomposition option that can be used with the Backward Euler solver that does an in-place LU decomposition, overwriting the A matrix with the L and U matrix data, and then uses this in the linear solve.

Acceptance Criteria

• An in-place LU decomposition and solve is available for use with the Backward Euler solver
• Tests of the Backward Euler solver with the in-place LU decomposition and solve are included for all the analytical test cases.

Ideas

• Collect the Jacobian non-zero elements, do the Diagonal Markowitz reordering, add any additional non-zero elements needed by the L and U matrices, and then create a jacobian/L/U combined sparse matrix in the State
• Overload the LinearSolver::Factor() and LinearSolver::Solve() functions with versions that accept a non-const reference to a single sparse matrix.