Scalable Nonlinear Compact Schemes

The Laboratory's main facility is outside Chicago, at 9700 South Cass Avenue, Argonne, Illinois 60439. For information about Argonne and its pioneering science and technology programs, see www.anl.gov. Solutions to hyperbolic conservation laws are often characterized by a large range of length scales as well as discontinuities. Standard nonlinear finite-difference schemes, such as the WENO schemes, yield non-oscillatory solutions but lack the spectral resolution required to model the relevant length scales. Linear compact schemes have a high spectral resolution; however, they suffer from spurious oscillations across discontinuities and sharp gradients. Weighted nonlinear compact schemes, such as the CRWENO scheme and the hybrid compact-WENO schemes, combine the non-oscillatory nature of the WENO schemes with the high spectral resolution of the compact schemes and are thus ideal for solutions with multiple length scales and discontinuities. One example of an application area is compressible, turbulent flows. The CRWENO scheme and the hybrid compact-WENO schemes have a nonlinear, solution-dependent left-hand side and therefore require the solution of banded systems of equations at each time-integration step or stage. Application of these schemes to multiprocessor simulations requires an efficient, scalable algorithm for the solution to the banded systems. Past efforts at implementing nonlinear compact schemes for parallel simulations suffer from one or more of the following drawbacks: parallelization-induced approximations and errors, complicated and inefficient scheduling of communication and computation, significant increase in the mathematical complexity of the banded systems solver, and high communication overhead. Therefore, these algorithms do not scale well for massively parallel simulations and are inefficient compared with the corresponding standard finite-difference schemes. In this work, we focus on compact schemes resulting in tridiagonal systems of equations, specifically the fifth-order CRWENO scheme. We propose a scalable implementation of the nonlinear compact schemes by implementing a parallel tridiagonal solver based on the partitioning/substructuring approach. We use an iterative solver for the reduced system of equations; however, we solve this system to machine zero accuracy to ensure that no parallelization errors are introduced. It is possible to achieve machine-zero convergence with few iterations because of the diagonal dominance of the system. The number of iterations is specified a priori instead of a norm-based exit criterion, and collective communications are avoided. The overall algorithm thus involves only point-to-point communication between neighboring processors. Our implementation of the tridiagonal solver differs from and avoids the drawbacks of past efforts in the following ways: it introduces no parallelization-related approximations (multiprocessor …