Optimal constraint projection for hyperbolic evolution systems

Techniques are developed for projecting the solutions of symmetric-hyperbolic evolution systems onto the constraint submanifold (the constraint-satisfying subset of the dynamical field space). These optimal projections map a field configuration to the nearest configuration in the constraint submanifold, where distances between configurations are measured with the natural metric on the space of dynamical fields. The construction and use of these projections are illustrated for a new representation of the scalar field equation that exhibits both bulk and boundary generated constraint violations. Numerical simulations on a black hole background show that bulk constraint violations cannot be controlled by constraint-preserving boundary conditions alone, but are effectively controlled by constraint projection. Simulations also show that constraint violations entering through boundaries cannot be controlled by constraint projection alone, but are controlled by constraint-preserving boundary conditions. Numerical solutions to the pathological scalar field system are shown to converge to solutions of a standard representation of the scalar field equation when constraint projection and constraint-preserving boundary conditions are used together.

Figure 1

Constraint violations for evolutions with ${\gamma }_1={\gamma }_2=0$, freezing boundary conditions, and no constraint projections. Plotted are radial resolutions $N_r=21,31,\dots ,61$; all curves lie on top of each other.

Figure 2

Convergence plot for the evolution presented in Fig. 1. Plotted are differences from the solution with radial resolution $N_r=81$.

Figure 3

Constraint violations for evolutions with ${\gamma }_1={\gamma }_2=0$, constraint-preserving boundary conditions, and no constraint projection. Solid curves are normalized by the quantity $||\nabla u(t)||$ while the dashed curves are normalized by $||\nabla u(0)||$. Decay of the normalization factor $||\nabla u(t)||$ rather than growth of the constraints causes the growth in the highest-resolution solid curves, which have constant roundoff-level constraint violations.

Figure 4

Convergence of evolutions shown in Fig. 3. Plotted are differences from the evolution with $N=81$, which is henceforth the reference solution $u_R$. Solid curves are normalized by $||u(t)||$ while the dashed curves are normalized by $||u(0)||$. Decay of the normalization factor $||u(t)||$ causes the growth in the highest-resolution solid curves, for which $||\delta u(t)||$ is constant at roundoff level.

Figure 5

Constraint violations for evolutions with ${\gamma }_2=-1/M$, constraint-preserving boundary conditions, and without constraint projection. The inset shows differences $||\delta u(t)||/||u(t)||$ from the reference solution of Fig. 4. The curves level off at late times because both numerator and denominator grow exponentially at the same rates.

Figure 6

Constraint violations for evolutions with ${\gamma }_1={\gamma }_2=0$, freezing boundary conditions, and a single constraint projection at $t=20M$ (with $\Lambda =2/M$). Points show $||C(t)||/||\nabla u(t)||$ after each time step. The inset plots the same data on a linear scale.

Figure 7

Convergence of evolutions shown in Fig. 6. Plotted are differences from the evolution with $N_r=81$.

Figure 8

Radial profiles of $⟨\psi {⟩}_{10}$ and $⟨C_i{C}^i{⟩}_{00}$ for the evolution of Fig. 6. The solid lines represent times $t/M=20,\dots ,25$. The dashed line represents the state just before the constraint projection at $t/M=20$. The arrows indicate the location of the nonsmoothness in $\psi$.

Figure 9

Differences between evolutions with time step $\Delta t$ and the reference solution $u_R$ (of Fig. 4) at fixed evolution times $t_0$. Evolutions use ${\gamma }_1={\gamma }_2=0$, freezing boundary conditions, and constraint projection with $\Lambda =2/M$ after each time step, $\Delta T=\Delta t$.

Figure 10

Constraint violations $||C(t)||/||\nabla u(t)||$ for evolutions with ${\gamma }_1=0$ and ${\gamma }_2=-1/M$, constraint-preserving boundary conditions, and constraint projection with $\Lambda =\sqrt{2}/M$ every $\Delta T=2M$. Inset shows the same data with finer time resolution.

Figure 11

Differences from the reference solution $u_R$ (of Fig. 4) for the evolutions shown in Fig. 10.

Figure 12

Differences $||\delta u(t)||/||u(t)||$ from the reference solution $u_R$ of Fig. 4 are plotted for different choices of $\Lambda$. Evolutions with ${\gamma }_1=0$ and ${\gamma }_2=-1/M$, constraint-preserving boundary conditions, constraint projection every $\Delta T=2M$.

Figure 13

Evolutions with ${\gamma }_1=0$ and ${\gamma }_2=-1/M$, constraint-preserving boundary conditions and constraint projection every $\Delta T$. Differences from the reference solution $u_R$ (of Fig. 4) at $t_0=100M$ for different choices of $\Delta T$ and $\Lambda$.

Figure 14

Solid curve (left axis) shows the ratio of time spent in elliptic solves to time spent in the hyperbolic evolution code. Dashed curve (right axis) shows the ratio of time required for one elliptic solve to the time for one evolution time step.