Prolongation

Aim

Implement more accurate prolongation in Carpet using the method described in Mongwane - Toward a Consistent Framework for High Order Mesh Refinement Schemes in Numerical Relativity.

Contributors

Eloisa Bentivegna, Roland Haas, Ian Hinder

Rationale

This should allow us to have 4th order accurate prolongation without buffer zones, instead of 2nd order accurate prolongation with buffer zones. This will allow much more flexibility in choosing refined regions, reduce the amount of prolongation, and avoid evolution of all the extra points in the buffer zones.

Useful references

• Mongwane - Toward a Consistent Framework for High Order Mesh Refinement Schemes in Numerical Relativity, Sec. 2.4.1
• Carpenter et al. - The Theoretical Accuracy of Runge--Kutta Time Discretizations for the Initial Boundary Value Problem: A Study of the Boundary Error
• Shen - Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations
• Gustafsson - The Convergence Rate for Difference Approximations to Mixed Initial Boundary Value Problems (one order lower at boundary)
• McCoquodale et al. - High-order finite-volume methods for hyperbolic conservation laws on mapped multiblock grids
• Problems in cosmology and numerical relativity (Bishop's PhD thesis)
• Pseudocode

Meeting - 07-Aug-2017

Discussion

• Read through the relevant section of the paper.
• Discussed whole method and brainstormed how we might implement it.
• Whiteboard

Tasks for next meeting

Eloisa:

• Set up test cases for space-independent Einstein solutions:
  • Convergence of unigrid - 4th order
  • Convergence of FMR, 2 level, with standard buffer zones. Expect 2nd order error at fixed t (due to order of time interpolation; three points, O(dt^3) local error, O(dt^2) error when integrated up to fixed t)
  • Convergence of FMR, 2 level, without buffer zones, and prolongating on every MoL substep. Expect 1nd order (Carpenter paper).
• Derive k4 if possible

Ian:

• Check we understand all the formulas and the algorithm
• Write out a complete description of what we want to implement

Roland:

• Look at MoL and Carpet and work out what sorts of functions need to be inserted/scheduled/adapted to implement the scheme

Sibo:

• For 1D python code, measure convergence in time for spatially-constant initial data with buffer zones - should be 2nd order at fixed t (I think)

Tomorrow we resync, to make sure we all understand all the parts.

Meeting - 08-Aug-2017

Reports

Ian:

• RK4 expressions we will use to prolongate.
• Algorithm notes

Eloisa:

• Cactus runs set up with EDS and 2-level FMR with 1. unigrid, 2. standard Carpet buffer zones, 3. no buffer zones
• See 4th order in unigrid and away from the refined region
• Refined regions no clear order
• Only 5 points in refined region; will be clearer with higher resolution

Tasks for next meeting

Eloisa:

• Add resolution to the FMR test runs in 1D to separate the different regions by the first common time step
• Finish derivation of the ks

Ian:

• Hash out a few Cactus function outlines for computing things, prolongating, etc, and schedules. Discuss tomorrow.

Roland:

• Look at the Cactus functions as well
• Think about the regridding issue
• Look at the python code again

Meeting - 10-Aug-2017

Reports

Eloisa:

• From three experiments (unigrid, no buffer zone, buffer zones): Got what we expected. Some nonconvergence starts at RBs and then propagates through the grid. This is with using both parameters needed to implement Berger Oliger in Carpet and MoL.
• Switched to RK2 so that everything should be 2nd order in the buffer zone simulation, but the order reduction should lead to 1st order at the RBs in the case with no buffer zones. However, the results, while close to 2nd order, were pretty identical between the two cases.
• So no need for buffer zones in this example. Standard Berger-Oliger is just as good.
• Now trying full 3D cosmology true-adaptive evolution with Berger-Oliger.
• Ian also interested in results of 1D convergence test with a spatially-dependent nonlinear solution (gauge wave). Will give parameters for initial data.
• Derived all expressions from Mongwane paper and agrees with them.

Roland:

• Worked on Python code. Fully tapered scheme leads to full 4th order accuracy for wave equation as expected.

Ian:

• Started implementing RK prolongation in Cactus. All implementation should be in MoL.
• Needed to fix a problem in Carpet relating to the "copy" prolongation operator.
• Currently have the ks prolongated, and the ks copied into the kTildes.
• Working on setting the RHS vars in the refinement boundaries from the kTildes (i.e. implementing the formulas from Mongwane).

Tasks for next meeting

Roland:

• Will test prolongation implemented by Ian actually works, with op_copy.

Eloisa:

• Continue to experiment with fully adaptive AMR simulation without buffer zones.

Ian:

• Continue with implementation of RK prolongation in MoL

Call with Bishop - 17-Aug-2017

• Most papers focused on RK methods. Other arguments for other methods. Errors cancel, so one order lower is ok.

• 1996 interpolants for rk formulae

• carpetner - wave eq in 2nd order, 3rd at boundary but reported 4th order.

• carpetner - jacobian term helps to get the nonlinear system. bishop thessis chapter 9 - put in jacobian by hand - for carpenter. for unigrid. gowdy wave. adm geodesic.

• order reduction - error propagates inwards! bdr has to be consistent.

• special case of sponge boundary

• difference is sponge applied at every substep

• tz applied at end of fine step. (maybe)

• need coarse grid approxes. soln for fine grid and coarse grid at same time - linear interpolation of grids. weight factor in linear interpolation gives smoothness - 3 transition zones - same thing. speed of prop depends on size of grid. mode speed changes abruptly cand causes refelctions. reduces reflections.

• sponge boundary can be as large as you need and use in restriction, but with transition zone, need to exclude transition from restriction - instabilities.

• interpolation is done using ks, ths, bs. can generate the ys

• also communicates the yn. so need 6 variables. just need yns, but keep other because it is solution.

• he has yn and yn+1 because he needs yn+1 anyway. we can use just yn from the coarse grid - need to prolongate one more variable.

Update - 17-Aug-2017

• Code implemented and tested for 2-level system
• Gauge wave and FTSS wave equation give 4th order convergence at low resolution, but convergence is not perfect at high resolution. Could be dropping to 3rd order there.
• Fully first order wave equation gives clean 4th order convergence at all resolutions
• Second derivatives need higher regularity of the solution; maybe they are spoiled by the sharp features introduced by RBs.
• Code extended to arbitrary number of refinement levels
• See numerical blowups with 3 levels with the wave equation. Suspect that we need to smooth out the sharp features at the refinement boundaries.
• Next step: implement transition zones from Bishop's paper

Use

• Use the rkprol branches of Carpet and MoL
• Set these parameters:
  • Carpet::use_buffer_zones = no
  • MoL::rk_prolongation = yes