Enhancing CFL barrier for transient systems

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.