A study of the temporal accuracy of a variety of first- and second-order time-integration methods applied to two-dimensional, multimaterial, nonequilibrium, radiation diffusion simulations is presented. These methods are categorized by their temporal order of accuracy, whether the algorithm includes operator splitting, and whether the algorithm includes linearizations. Results are presented that simultaneously measure accuracy and efficiency of the different methods on two different test problems. The two test problems are designed to represent an easy problem, where different approximations may be accurate, and a hard test problem that will stress the different solution algorithms. Results show the importance of being second-order accurate in time and the importance of time-step control.