• Accurate parallel integration of large sparse systems of differential equations, D. Estep and R. Williams, Mathematical Models and Methods in Applied Sciences 6 (1996), 535-568
  We consider the implementation of a posteriori error analysis of numerical methods for large dimensional systems of ordinary differential equations and apply the methods to analyze the behavior the bistable problem. We address parallelization issues in particular.
• The computability of the Lorenz system, D. Estep and C. Johnson , Mathematical Models and Methods in Applied Sciences 8 (1998), 1277-1305.
  We perform a numerical analysis of the chaotic Lorenz and forced Duffing problems using accurate computation based on adaptive error control. We explain how detailed information about the dynamical behavior of these systems can be obtained by accurate short-time computations despite the accumulation of error associated to chaotic behavior.
• Computational error estimation and adaptive mesh refinement for a finite element solution of launch vehicle trajectory problems, D. Estep, D. Hodges and M. Warner, SIAM Journal on Scientific Computing 21 (2000), 1609-1631 (electronic)
  In this paper, we consider a new kind of discontinuous Galerkin finite element method for first order initial-final value problems arising from optimal control of launch vehicles. In particular, we derive an a posteriori error estimate for this method that is subsequently implemented to provide a computational error estimate used for adaptive error control. We discuss several practical issues in the implementation of the computational error estimate as well. We test the theory on a real-life trajectory problem and find that the estimates are reliable and accurate while the adaptive error control leads to a significant gain in efficiency. The resulting method is sufficiently fast to allow for real-time control of a launch vehicle.
• The solution of a launch vehicle trajectory problem by an adaptive finite element method, D. Estep, D. H. Hodges, M. Warner, Computer Methods in Applied Mechanics and Engineering, 190 (2001), 4677-4690.
  We present a simplified analysis of a new kind of discontinuous Galerkin finite element method for the solution of an optimal control problem for launch vehicles and conduct extensive tests of the accuracy and reliability of the computational error estimate.
• Analysis of shear layers in a fluid with temperature-dependent viscosity, D. Estep, S. Verduyn Lunel, and R. Williams, Journal on Computational Physics 173 (2001), 17-60.
  We investigate the behavior of a singular system of reaction diffusion equations modeling the shear flow in a fluid with a viscosity that decreases as temperature increases both analytically and numerically. This issue is a long-standing question about whether or not the solutions exhibit finite time blowup. The numerical investigation uses an adaptive finite element method that preserves several dynamical properties of the underlying system. We also introduce a new kind of asymptotic analysis to supplement the numerical results. Our analysis covers behavior on a extremely wide range of scales and produces remarkable agreement between the analytic and numeric results.