Mathematical foundations of LEAP

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Abstract
As part of the ORNL energy model validation program an analysis was made of the mathematical structure of the LEAP energy modeling code. Emphasis was placed on understanding the equations which the code attempts to solve and the iterative numerical algorithm used to achieve a converged solution. The equations were found to be highly nonlinear, similar in form to those found in equilibrium boundary value problems. A combination of a standard nonlinear Gauss-Seidel approach and a modification of Newton's method was used as the basis of an iterative algorithm to find local solutions to the equilibrium equations. Several weaknesses in this approach are discussed, mainly those dealing with the computaton of relaxation coefficients in the Newton's method segment of the algorithm. The chief failing of the method as employed in LEAP is the use of a diagonal Jacobian matrix for Newton's method with an ad hoc scheme for calculating the diagonal matrix elements. Most problems encountered had large off-diagonal elements leading to a characteristic oscillatory break-down in convergence. It was concluded that the LEAP algorithm cannot be used to achieve convergence in an automated fashion for any general problem. In practice hand intervention is needed in most cases. Recommendations aremore » made to approximate certain key elements of the Jacobian matrix to improve the general usefulness of the algorithm. Analytic expressions for the approximate elements are derived so as to facilitate implementation in LEAP. Other aspects of the numerical procedures are discussed, and recommendations for their improvement are also made.« less

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