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Variable-implicit solver and dualtime_advance() exposure

James Sutherland requested to merge variable-implicit into master
  • Add variable-implicit implicit BDF method, which uses the diagonal of the Jacobian matrix
  • Add corresponding sensitivities to ODE test cases and the general BDF residual expression
  • Modify implicit integrators to expose internals of each physical time step
  • All dual time tests now use the new framework, running both the physical and dual time loops
  • All tests but the heat equation use an initialization tree, required for sensitivity propagation
  • Modify integrators to use expressions for the physical and dual time step
  • Modify regression test of fixed-point BDF-1 to check convergence rate
  • Add regression test for variable-implicit that checks both solution accuracy and convergence rate
  • Rename several dual time integrator members and methods
  • Add Matlab test for exp decay problem with fixed-point and variable-implicit

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