Week 2/6

1- Read the paper “Time accurate local time stepping for the unsteady shallow water equations”. There is mesh adaptivity in their tests where larger cells in the boundaries and finer cells are used in the center (page 790). The maximum achieved speed up is about 2x when compared to global time stepping (GTS), but accuracy of solution with LTS improved in some cases as well.

2- Read the sections related to time stepping adaptivity of the paper about integrating Peano in pyclaw.

Week 9/6

1. I had looked into some details of the paper “A DIRECT MULTIGRID POISSON SOLVER FOR OCT-TREE ADAPTIVE MESHES” to understand a number of comments I received from Prof.Donna Calhoun regarding elliptic solvers on AMR.

2. I’ve scanned some papers that uses AMR. Mainly references 4,5,6,16 from the paper about peano and pyclaw. As far as I understood no one uses sophisticated approach to determine the local time step (the level of refinement in time is the same as the level of refinement of space). I tried to look for papers that combine “advanced” LTS with AMR but I could not find.

3. I’ve looked for some references about Parareal algorithm and started reading “The Parareal Algorithm: A survey of present work”.

Week 16/6

1. Continue reading the report “The Parareal Algorithm: A survey of present work”. Key points in this report:
   • The Parareal algorithm can be thought of applying the idea of domain decomposition in time “time decomposition”. However, unlike domain decomposition, time decomposition done by Parareal has no value in serial computations.
   • This algorithm has stability issues in case the system matrix has pure imaginary eigenvalues (or where the imaginary part is much larger than the real part).
   • The algorithm uses a coarse time propagator and a fine time propagator. The first one works sequentially through the time domain (to define initial point for each subdomain - processor) then the fine time propagator works sequentially in each processor where each processor evolve a solution through a time subdomain simultaneously. A sequential corrector-predector step is then performed. To converge, several iterations of those operations are usually required.
   • It is analogues to the conjugate gradient method in the sense that after \(N-1\) iteration (\(N\) is the number of subdomains), the Parareal solution will converge exactly to the serial solution. The point is, it may converge in much less iterations achieving significant speedup.
   • Assuming the cost of the coarse propagator over the whole domain is equal to the cost of the fine propagator in one domain, speed up will be achieved only if the solution converges in \(k<\frac{N}{2}\) iterations. Speeding up the coarse propagator (e.g. applying it to a coarser grid) will enhance the speed up.
   • The parareal solution is never more accurate than the corresponding serial solution.
   • Two different solvers can be used for the coarse and the fine propagator (e.g. implicit solver can be used for the coarse propagator allowing large time steps while explicit solver can be used for the fine propagator). Single-step solvers are favorable.
   • If the coarse propagator has order of accuracy \(m\ge 1\), the order of accuracy of the Parareal algorithm is \((m+1)(k+1)\) where \(k\) is the number of iterations. Note that we are considering the convergence to the corresponding serial solution, not the exact solution.
   • The maximum parallel efficiency of the Parareal algorithm is \(E=\frac{1}{2}\). However, by introducing a “multi-step system” the efficiency can be very close to 1 (this idea is not discussed further in this report).
   • The algorithm have been applied to several problems. It is mentioned that this algorithm gave good results (some results are only simulated) for linear and nonlinear parabolic problems, Molecular dynamics, optimal control for PDEs, reservoirs simulations, unsteady flow and Navier-stokes for small Reynolds numbers.
   • More research can be done in reducing the sequential part of the algorithm. Also, research is needed to solve the difficulties related to the stability of this algorithm when applied to problems like Navier-stoks with high Reynolds numbers, hyperbolic equations and conservation laws and structural dynamic vibration problems.

Weeks 23/6 - 30/6

1. Wrote preliminary outline for the thesis proposal, found here (limited access).

2. Looked for candidate papers about parallel time integration to read.
   • M. J. Gander, “Analysis of the parareal algorithm applied to hyperbolic problems using characteristics”
   • R. Croce, D. Ruprecht, and R. Krause, “Parallel-in-Space-and-Time Simulation of the Three- Dimensional , Unsteady Navier-Stokes Equations for Incompressible Flow,” 2012.
   • C. Farhat and M. Chandesris, “Time-decomposed parallel time-integrators: theory and feasibility studies for fluid, structure, and fluid-structure applications,” International Journal for Numerical Methods in Engineering, vol. 58, no. 9.
   • G. Bal, “Parallelization in time of (stochastic) ordinary differential equations,” (which propose parallel in time algorithm with efficiency close to 1)

3. Read the paper “Spectral Deferred Correction Methods for Ordinary Differential Equations”, by A. Dutt, L. Greengard and V. Rokhlin. My summary of the paper can be found here.

Week 7/7

1. Reading the paper “Analysis of the Parareal Algorithm Applied to Hyperbolic Problems Using Characteristics”. My summary of the paper can be found here.

2. Reading the paper “Toward an Efficient Parallel in Time Method for Partial Differential Equations”. (Incomplete)

3. Added some more references and few thoughts in the proposal

Weeks 14-21-28/7

Working on the proposal

Weeks 4/8 - 11/8

1. Working on the proposal
2. My summary of the chapter “A Time-Parallel Implicit Methodology for the Near-Real-Time Solution of Systems of Linear Oscillators” by Cortial and Farhat can be found here

Week 18/8

Eid vacation

Week 25/8