=================================================================== #ECSG - A package for 3D anisotropic elastic wave simulations based on a stable composite staggered grid finite difference time domain numerical scheme

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Why the composite grid?

It is well known that finite difference scheme suffers from grid dispersion. To minimize this numerical aritifact, we need to sample each wavelength by at least a certain number of grid points. When the simulation domain contains a low velocity zone, for example the shallow region, using the classical uniform grid scheme (same mesh size over the entire domain) would require a very fine mesh. In contrast, a composite grid scheme, which uses fine mesh for low velocity region and coarse mesh for high velocity region, needs less grid points to simulate the same region, so it helps to reduce the memory and computing cost of the simulation.

Why we need stability?

In numerical analysis, numerical stability is always an important property we are looking for when we design a numerical scheme. Stability means a perturbation in the initial solution (e.g., machine error) will not blow up over time. Lax-Richtmyer theorem claims that for a linear consistent numerical scheme (e.g., finite difference scheme based on the wave equation), stability is a sufficient and necessary condition for convergence, which measures how close the numerical solution will be to the true solution if we keep reducing the mesh size. Because of these reasons, we need a stable numerical scheme.

What is energy method and how we use the energy method to achieve numerical stability for this composite staggered grid FDTD scheme based on the elastic wave equation?

Energy method is a powerful technique to analyze and derive stability conditions. It starts by formulating an energy of the numerical solution, which is often a bilinear semidefinite form of the numerical solution. Note the energy is not the same energy as the one defined in physics, but often this energy is formulated based on the physics definition. For example, we formulate the energy of the wave equation solution based on the kinetic and strain energy. If the energy is conserved over time, and if the time step is chosen to be sufficiently small such that the energy is equivalent to the L2 norm of the solution, then the scheme is stable.

As opposed to von Neumann stability analysis which works on only periodic boundary condition and gives only necessary stability condition, energy method can work on non-periodic boundary condition, e.g., rigid, free surface boundary condition, and it can yield the sufficient stability condition.

After formulating the energy for the numerical solution of the second-order anisotropic elastic wave equation on a uniform staggered grid, we compute the energy on a composite grid by summing up energies on each individual uniform grid. We find that in order to conserve the energy on this composite grid, the unknown ghost data near the grid refinement interface, which are built to compute interface field data, have to satisfy a certain condition, i.e., a set of linear equations with those ghost data as unknowns. On the other hand, the transmission condition across the interface imposes another set of linear equations on the ghost data. These two sets of linear equations formulate three independent linear equation systems to solve for the ghost data. In our code, we use both direct solvers and iterative solver (Jacobi Method) to solve these linear equation systems. To make the energy on the composite grid to be equivalent to the L2 norm of the solution, we derived a sufficient stability condition which yields an upper bound for the time step.

What is inside this code package?

This package is built to verify numerical stability and convergence of 3D elastic wave simulations based on this stable compostie staggered grid FDTD scheme.

1. This package is able to perform both single grid and multiple (>=2) grids schemes with energy-conserving interpolation implemented at grid refinement interfaces to ensure numerical stability.
2. This package relies on Madagascar, OpenMP and scons, so have them installed or loaded before compiling the code. You may need to modify paths when running tests.
3. Currently the scheme is based on FDTD of second order in time and space. Free surface boundary condition is implemented on top, and rigid (Dirichlet) boundary condition is implemented on other three sides of the rectangular domain.

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Structure

• SConstruct-local/SConstruct-tacc: python scripts for code compilation on OSX and TACC/Stampede respectively. To compile this package, copy either one of the scripts to SConstruct, and then run scons.
• main.cpp: driver of this program.
• init.hpp: set up precision macros.
• grid.hpp: A C struct defines grid.
• field.hpp/field.cpp: A C++ class defines simulation fields and related operations.
• array.hpp/array.cpp: A C++ class defines multi-dimensional matrices.
• sim_utils.hpp: list of simulation function signatures with functions defined in:
  • init_del_par_field.cpp: functions initialize and delete parameter fields.
  • init_del_sim.cpp: functions initialize and delete simulation fields
  • init_del_ghost_container.cpp: functions initialize and delete interface ghost data.
  • init_del_fd_coeff.cpp: functions initialize and delete finite difference coefficients.
  • init_src.cpp: functions initialize a Ricker source.
  • init_del_rec_trace: functions initialize and delete a set of receiver traces.
  • init_del_movie.cpp: functions initlalize and delete rsf movies.
  • init_del_res_trace.cpp: functions initialize and delete residue traces
  • update_*.cpp: functions update velocity/stresses/ghost data at grid interface.
  • compute_energy.cpp/compute_res.cpp: functions compute energy and residue at a composite grid.
  • print.cpp: functions print receiver data/residues/movies to rsf files.
• test_utils.hpp: list of test function signatures with functions defined in:
  • test_sbp.cpp: functions test SBP property.
• tests: test scripts

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Future Work

• Extend this composite grid scheme to deal with fluid-solid interface.
• Extend this composite grid scheme to high order FDTD.
• MPI parallelization.