Adaptive mesh refinement for coupled elliptic-hyperbolic systems

Abstract

We present a modification to the Berger and Oliger adaptive mesh refinement algorithm designed to solve systems of coupled, non-linear, hyperbolic and elliptic partial differential equations. Such systems typically arise during constrained evolution of the field equations of general relativity. The novel aspect of this algorithm is a technique of "extrapolation and delayed solution" used to deal with the non-local nature of the solution of the elliptic equations, driven by dynamical sources, within the usual Berger and Oliger time-stepping framework. We show empirical results demonstrating the effectiveness of this technique in axisymmetric gravitational collapse simulations, and further demonstrate that the solution time scales approximately linearly with problem size. We also describe several other details of the code, including truncation error estimation using a self-shadow hierarchy, and the refinement-boundary interpolation operators that are used to help suppress spurious highfrequency solution components ("noise").

Evolution to a smooth universe in an ekpyrotic contracting phase withw>1

“…The elliptic slicing condition is incorporated into the Berger and Oliger time-stepping algorithm using the method described in [10]. Such modifications are necessary to take advantage of time-subcycling, however here we find that we can evolve the system without timesubcycling yet keep the time step equal to that of the coarsest level in the hierarchy.…”

“…We discretize the equations using second-order accurate finite difference techniques, and solve them with a variant of the Berger and Oliger [9] adaptive mesh refinement (AMR) algorithm for coupled elliptic-hyperbolic equations [10]. We find smooth regions that are scalar field dominated in which the scalar field (kinetic plus potential energy density) component behaves like a fluid with w ≫ 1, and also regions where the scalar field kinetic energy dominates over the potential energy and the scalar field behaves like a fluid with w = 1.…”

Calibration of moving puncture simulations

“…For example, quadratic interpolation for ICN and a first-order in time, secondorder in space formulation can lead to a drop of convergence order and instabilities; see Schnetter et al [64]. Other authors report success with different variants of time interpolation, e.g., [65,66].…”

Constrained evolution in axisymmetry and the gravitational collapse of prolate Brill waves

“…Figure 2 confirms that the code is approximately second-order convergent. (Occasional values Q u > 4 are not uncommon in similar numerical schemes [4,29]. )…”

“…Including elliptic equations in this approach is rather complicated. A solution with numerical relativity applications in mind was suggested by Pretorius and Choptuik [29], and we shall use their algorithm here, with minor modifications due to the fact that our grids are cell centred rather than vertex centred. The key idea of the algorithm is that solution of the elliptic equations on coarse grids is deferred until all finer grids have reached the same time; meanwhile the elliptic unknowns are linearly extrapolated in time and only the evolution equations are solved.…”