In the hydrodynamic regime, the evolution of a stochastic lattice gas with symmetric hopping rules is described by a diffusion equation with density-dependent diffusion coefficient. In practice, even when the equilibrium properties of a lattice gas are analytically known, the diffusion coefficient cannot be explicitly computed, except when a lattice gas additionally satisfies the "gradient condition", e.g. the diffusion coefficients of the simple exclusion process and non-interacting random walks are exactly identical to their hopping rates. We develop a procedure to obtain systematic analytical approximations for the diffusion coefficient in non-gradient lattice gases with known equilibrium. The method relies on a variational formula found by Varadhan and Spohn. Restricting the variational formula to finite-dimensional sub-spaces allows one to perform the minimization and gives upper bounds for the diffusion coefficient. We demonstrate the procedure for the following two models; one-dimensional generalized exclusion processes (GEPs) in which each site can accommodate at most two particles, and the Kob-Andersen model on the square lattice, which is classified into kinetically-constrained gas . The prediction of the diffusion coefficient depends on the domain ("shape") of test functions. The smallest shapes give approximations which coincide with the mean-field theory, but by using larger shapes we obtain better upper bounds. For the GEPs, our analytical predictions provide upper bounds which are very close to simulation results throughout the entire density range.
For the KA model, we also find improved upper bounds when the density is small. By combining the variational method with a perturbation approach, we discuss the asymptotic behavior of the diffusion coefficient in the high density limit.