Abstract:
A number of scientific and engineering systems are governed by PDEs which have solutions comprising wide range of spatial and temporal scales. The accompanying details can only be captured
by high-fidelity simulations on high performance computational systems. Although, advances in
computing technology have made it possible to carry out intensive simulations on massively parallel
computers, the presence of order of magnitude gaps in spatial and temporal scales is a major obstacle
to sustained performance for a number of scientific codes. Thus, the development of PDE solvers
which can bridge these order of magnitude gaps in spatial and temporal scales is a major challenge for
computational physics and forms a central topic for high-performance computing. In this thesis, we
have presented a general methodology to analyze and derive explicit schemes that overcome the issue
of small time-steps by allowing some tunable level of asynchrony i.e. using spatial data from present
and previous time levels for computing derivatives.
The concept relies on finite differences to approximate derivatives using values of the function
from neighboring points and the realization that wider stencils have better stability. The performance
of explicit solvers is stalled due to the time-step restrictions imposed by the stability criteria which
is more severe for a problem with higher number of degrees of freedom. In few recent works, it
was shown that the stability limit for the explicit solver of the diffusion equation was doubled in the
delayed difference scheme where computations can proceed using values from past time levels. In this
work, we highlight a checkerboard instability in the formulation of the delayed difference scheme for
1D diffusion equation. In order to overcome this issue, we consider a general scheme that is weighted
between the delayed and the conventional explicit schemes. We have established via stability analysis
that an optimum value of the weighting factor exists such that the critical CFL number obtained is
larger than the original delayed difference scheme. These schemes are referred as weighted difference
schemes.
The multidimensional extension of weighted differencing approach for diffusion equation is
accomplished using isotropic differential operators to compute the spatial derivatives. By analyzing
in detail the effective differential equation, we demonstrate the consistency and order of accuracy
of weighted scheme. We have provided a general framework in which stability of schemes can be
accessed by analyzing the eigen value spectra of the diffusion operator for different discretization
procedure namely central difference and isotropic operators. Fourier analysis of the diffusion operator
revealed that isotropic discretization, ∇·∇, alleviates the dependency of CFL on dimension of problem as opposed to the conventional central difference discretization. The use of isotropic differential
operators along with weighted combination of delayed and non-delayed difference schemes has shown
significant improvement in CFL criteria for multi-dimensional problems.
In order to access the computational efficiency of weighted schemes, we have performed a series
of numerical experiments for transient and steady computations and compared the performance
against standard explicit time marching schemes and iterative solvers. Theoretical predictions on the
accuracy and convergence rates of weighted schemes have been compared to numerical experiments
for transient and steady problems. Good agreement has been found across different initial and
boundary conditions. We have also performed weak and strong scaling studies for the parallel version
of the code on Intel KNL architecture. The work presented here provides a strong foundation for
deriving schemes which inherit the simplicity and parallel efficiency of the Jacobi method while
retaining the fast convergence property of the Gauss-Seidel method.