Chaotic Method of Manufactured Solutions: Lorenz 63

I quickly brushed over the method of manufactured solutions (MMS) aspects in the previous post on Lorenz 63, you might wonder why. Well, it turns out that applying the MMS to chaotic systems includes an additional consideration that is not normally of concern. Here is the ’standard’ sort of checklist for choosing manufactured solutions (taken from Verification of Computer Codes in Computational Science and Engineering, emphasis mine):

1. Manufactured solutions should be sufficiently smooth on the problem domain so that the theoretical order-of-accuracy can be matched by the observed order-of-accuracy obtained from the test. A manufactured solution that is not sufficiently smooth may decrease the observed order-of-accuracy (however, see # 7 in this list).
2. The solution should be general enough that it exercises every term in the governing equation. For example, one should not choose temperature T in the unsteady heat equation to be independent of time. If the governing equation contains spatial cross-derivatives, make sure the manufactured solution has a non-zero cross derivative.
3. The solution should have a sufficient number of nontrivial derivatives. For example, if the code that solves the heat conduction equation is second order in space, picking T in the heat equation to be a linear function of time and space will not provide a sufficient test because the discretization error would be zero (to within round-off) even on coarse grids.
4. Solution derivatives should be bounded by a small constant. This ensures that the solution is not a strongly varying function of space or time or both. If this guideline is not met, then one may not be able to demonstrate the required asymptotic order-of-accuracy using practical grid sizes. Usually, the free constants that are part of the manufactured solution can be selected to meet this guideline.
5. The manufactured solution should not prevent the code from running successfully to completion during testing. Robustness issues are not a part of code order verification. For example, if the code (explicitly or implicitly) assumes the solution is positive, make sure the manufactured solution is positive; or if the heat conduction code expects time units of seconds, do not give it a solution whose units are nanoseconds.
6. Manufactured solutions should be composed of simple analytical functions like polynomials, trigonometric, or exponential functions so that the exact solution can be conveniently and accurately computed. An exact solution composed of infinite series or integrals of functions with singularities is not convenient.
7. The solution should be constructed in such a manner that the differential operators in the governing equations make sense. For example, in the heat conduction equation, the flux is required to be differentiable. Therefore, if one desires to test the code for the case of discontinuous thermal conductivity, then the manufactured solution for temperature must be constructed in such a way that the flux is differentiable. The resultant manufactured solution for temperature is nondifferentiable.
8. Avoid manufactured solutions that grow exponentially in time to avoid confusion with numerical instability.

As emphasized in # 4 we often have to use the free constants in our manufactured solution to bound the derivatives in the domain. For chaotic systems we’ll need to use not only the free constants, but the initial conditions as well, to achieve another end: ensure that the chosen solution is in the convergent region at all times. This is a sufficient but not necessary condition for the source term to be in the basin of entrainment. When the source term is in the basin of entrainment then we can expect to be able to verify the ordered convergence of our implementation as we would in the non-chaotic case. Otherwises we can expect the truncation error to eventually lead to an arbitrarily large departure from the manufactured solution. This terminology comes from the nonlinear dynamics and control literature [1] [3]. A worked example for the Lorenz ’63 system showing the boundary of the convergent region is in Chapter 6 of [2].

The convergent region is defined by the region of phase space for which the eigenvalues of the system Jacobian has negative real parts. As in any other application of the MMS, we start by choosing some convenient solution consistent with the desiderata enumerated above. Trigonometric functions are convenient in this case:

(1)

(2)

(3)

Now we need the Jacobian of the system (and then its eigenvalues), since our governing equations are definied in Maxima, that is easy enough:

/* Need the Jacobian of the system to find the ’convergent region’ so 
   we can select a useful IC/amplitude for our manufactured solutions */ 
sys_eqns : [eqn_1, eqn_2, eqn_3] $ 
J_cr : zeromatrix(3,3) $ 
for j : 1 thru 3 do 
( for i : 1 thru 3 do 
  ( 
    J_cr[i,j] : diff(rhs(sys_eqns[i]), unknowns[j]) 
    ) 
  ) $ 
[J_cr_eigvals, J_cr_eigmult] : eigenvalues(J_cr) $ 
J_cr_eigvals : fullratsimp(J_cr_eigvals) $ $

As before these expressions are output in Fortran90 format for compilation. Here are the eigenvalues in the complex plane for b_x = 1.0, b_y = -1.0, b_z = 10.0.

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Figure 1: Jacobian eigenvalues for manufactured solution, bx = 1.0, by = -1.0, bz = 10.0

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We can immediately see the problem from Figure 1: our manufactured solution is in the region of phase space where one of the Jacobian’s eigenvalues has a positive real part. We can adjust the initial conditions of our solution to get all of our eigenvalues on the left half-plane.

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Figure 2: Jacobian eigenvalues for manufactured solution, bx = 10.0, by = -10.0, bz = 40.0

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Being limited to only certain portions of the phase space for conducting verification is fine, because a single well-designed manufactured solution verifies the correctness of the entire implementation (as long as all of the terms are activated). This is the same reason that it is often emphasized in the V&V literature that there is no need for the manufactured solution to be physically meaningful.

References

[1]   Jackson, E.A., “Controls of dynamic flows with attractors,” Physics Review A, Vol. 44, Is. 8, 1991.

[2]   Wu, W., “Analytical and Numerical Methods Applied to Nonlinear Vessel Dynamics and Code Verification for Chaotic Systems,” Disertation, Virginia Polytechnic Institute and State University, Dec, 2009.

[3]   W. Wu, L.S. McCue, and C.J. Roy. “The method of manufactured solutions applied to chaotic systems,” Nonlinear Dynamics, 2009, under review.