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tokens: We see that the Strang CTU algorithm achieves second order for most variables (as expected), with some quantities converging to almost third order (for smooth flows, the PPM algorithm approaches third-order accuracy in space), while having difficulty with . For SDC-2, we see second-order convergence in all the variables, including . Finally, for SDC-4, all of the variables converge at rates of ∼3.8–3.9, demonstrating the fourth-order accuracy expected for the method. This test shows that the SDC algorithm can achieve fourth-order convergence for reactive hydrodynamics problems with astrophysical networks. Figure 7. Thermodynamic, dynamic, and nuclear state at t = 0.06 s for the reacting hydrodynamics test run with SDC-4 using 128 2 cells. Table 6 Convergence ( L 1 Norm) for the Reacting Convergence Problem with the Strang CTU Solver Table 7 Convergence ( L 1 Norm) for the Reacting Convergence Problem with the SDC-2 Solver Table 8 Convergence ( L 1 Norm) for the Reacting Convergence Problem with the SDC-4 Solver 5.6. Burning Buoyant Bubble Our final convergence test problem considers a hydrostatic atmosphere with a temperature perturbuation. 11 11 This problem setup is available in Castro as Exec/reacting _ tests/bubble _ convergence . Buoyancy causes the perturbation to rise (and eventually roll up in the nonlinear phase). The presence of reactions prevents the bubble from fizzling out, keeping it buoyant via the heat deposition. In addition to looking at the convergence rate of the numerical solutions, we also consider how well we maintain hydrostatic equilibrium in an undisturbed hydrostatic atmosphere. We create an initial atmospheric model that is isentropic and in hydrostatic equilibrium by integrating the system where the relation ρ ( p , s ) is provided by our equation of state. We take gravity, g , to be constant and the composition to be uniform throughout the atmosphere (pure helium, with the other nuclei mass fractions set to the small value 10 −8 ). To integrate this system, we specify the conditions at the base of the atmosphere, which we take to be the lower domain boundary (not the center of the bottom-most cell). We specify and , and get and through the general stellar equation of state. We integrate this system using fourth-order Runge–Kutta, using a step size of Δ x /2 to get from the bottom of the domain to the first cell center, and then a step size of Δ x to integrate to each of the remaining cell centers vertically in the domain. The initial conditions are then converted to cell averages using the same transformation discussed earlier in the paper. Note that the hydrostatic model is generated specifically for the resolution of the problem, and as such, the initial atmosphere converges with fourth-order accuracy. For the boundary conditions, we use periodic conditions on the sides and reflecting boundary conditions at the top and bottom. Table 9 lists the problem setup parameters. Table 9 Hydrostatic Atmosphere