I am developing a MPI based parallel numerical solver for a Laplace equation subjected to Dirichlet Boundary condition using Finite Difference method.I have taken a square computational domain with a grid size of 1024 x 1024. My objective is to evaluate the Scalability of the developed solver.
I have tested the code using 4,8,16,32,64,128 and 256 processors on a HPC facility having 8 nodes with each node consisting of 48 cores. Until 128 processors I could find a linear scalability in the computational time.(i.e between two consecutive cores say 4 and 8, 8 and 16, 16 and 32, 32 and 64 the solution converges and attains the set L2 and L-infinity true error norms values in the order of 10^-7 and 10^-8 without any issues and also the computational time reduces by half. whereas when i run the code using 128 and 256 cores the error norm values are not following a similar trend as that for 64 cores. The convergence rate drastically reduces say for 128 processors the error norm value is not reducing beyond 10^-6 and it takes more than half the computational time for 64 cores.Similarly with 256 cores the convergence rate is very poor and the computational time is very high.So, I request the experts in this field to help me to resolve this issue.