Narrow Results

This paper presents a hybrid Godunov method for three-dimensional radiation hydrodynamics. The multidimensional technique outlined in this paper is an extension of the one-dimensional method that was developed by Sekora and Stone 2009, 2010. The earlier one-dimensional technique was shown to preserve certain asymptotic limits and be uniformly well behaved from the photon free streaming (hyperbolic) limit through the weak equilibrium diffusion (parabolic) limit and to the strong equilibrium diffusion (hyperbolic) limit. This paper gives the algorithmic details for constructing a multidimensional method. A future paper will present numerical tests that demonstrate the robustness of the computational technique across a wide-range of parameter space.

The goal of this study is to develop a methodology for tracking the distribution of filtered solute in mathematical models of the urine concentrating mechanism. Investigation of intrarenal solute distribution, and its cycling by way of countercurrent exchange and preferential tubular interactions, may yield new insights into fundamental principles of concentrating mechanism function. Our method is implemented in a dynamic formulation of a central core model that represents renal tubules in both the cortex and the medulla. Axial solute diffusion is represented in intratubular flows and in the central core. By representing the fate of solute originally belonging to a marked bolus, we obtain the distribution of that solute as a function of time. In addition, we characterize the residence time of that solute by computing the portion of that solute remaining in the model system as a function of time. Because precise mass conservation is of particular importance in solute tracking, our numerical approach is based on the second-order Godunov method, which, by construction, is mass-conserving and accurately represents steep gradients and discontinuities in solute concentrations and tubular properties.