We describe a method for analyzing the structure of a system of nonlinear integro-differential–algebraic equations (IDAEs) that generalizes the Σ -method for the structural analysis of differential–algebraic equations. The method is based on the sparsity pattern of the IDAE and the ν -smoothing property of a Volterra integral operator. It determines which equations and how many times they need to be differentiated to determine the index, and it reveals the hidden constraints and compatibility conditions in order to prove the existence of a solution. The success of the Σ -method is indicated by the non-singularity of a certain Jacobian matrix. Although it is likely the Σ -method can be directly applied with success to many problems of practical interest, it can fail on some solvable IDAEs. Accordingly, we also present two techniques for addressing these failures.

Abstract The intent of this paper is to encourage improved numerical implementation of land models. Our contributions in this paper are two-fold. First, we present a unified framework to formulate and implement land model equations. We separate the representation of physical processes from their numerical solution, enabling the use of established robust numerical methods to solve the model equations. Second, we introduce a set of synthetic test cases (the laugh tests) to evaluate the numerical implementation of land models. The test cases include storage and transmission of water in soils, lateral sub-surface flow, coupled hydrological and thermodynamic processes in snow, and cryosuction processes in soil. We consider synthetic test cases as “laugh tests” for land models because they provide the most rudimentary test of model capabilities. The laugh tests presented in this paper are all solved with the Structure for Unifying Multiple Modeling Alternatives model (SUMMA) implemented using the SUite of Nonlinear and DIfferential/Algebraic equation Solvers (SUNDIALS). The numerical simulations from SUMMA/SUNDIALS are compared against (1) solutions to the synthetic test cases from other models documented in the peer-reviewed literature; (2) analytical solutions; and (3) observations made in laboratory experiments. In all cases, the numerical simulations are similar to the benchmarks, building confidence in the numerical model implementation. We posit that some land models may have difficulty in solving these benchmark problems. Dedicating more effort to solving synthetic test cases is critical in order to build confidence in the numerical implementation of land models.