Debojyoti Ghosh

COMPACT RECONSTRUCTION SCHEMES WITH WEIGHTED ENO LIMITING FOR HYPERBOLIC CONSERVATION LAWS

The simulation of turbulent compressible flows requires an algorithm with high ac- curacy and spectral resolution to capture different length scales, as well as nonoscillatory behavior across discontinuities like shock waves. Compact schemes have the desired resolution properties and thus, coupled with a nonoscillatory limiter, are ideal candidates for the numerical solution of such flows. A class of compact-reconstruction weighted essentially non-oscillatory CRWENO schemes is presented in this paper where lower order compact stencils are identified at each interface and combined using the WENO weights. This yields a higher order compact scheme for smooth solu- tions with superior resolution and lower truncation errors, compared to the WENO schemes. Across discontinuities, the scheme reduces to a lower order nonoscillatory compact scheme by excluding stencils containing the discontinuity. The schemes are analyzed for scalar conservation laws in terms of accuracy, convergence, and computational expense, and extended to the Euler equations of fluid dynamics. The scalar reconstruction is applied to the conserved and characteristic variables. Nu- merical test cases are presented that show the benefits of these schemes over the traditional WENO schemes.

Weighted Non-linear Compact Schemes for the Direct Numerical Simulation of Compressible, Turbulent Flows

A new class of compact-reconstruction weighted essentially non-oscillatory (CRWENO) schemes were introduced (Ghosh and Baeder in SIAM J Sci Comput 34(3): A1678–A1706, 2012) with high spectral resolution and essentially non-oscillatory behavior across discontinuities. The CRWENO schemes use solution-dependent weights to combine lower-order compact interpolation schemes and yield a high-order compact scheme for smooth solutions and a non-oscillatory compact scheme near discontinuities. The new schemes result in lower absolute errors, and improved resolution of discontinuities and smaller length scales, compared to the weighted essentially non-oscillatory (WENO) scheme of the same order of convergence. Several improvements to the smoothness-dependent weights, proposed in the literature in the context of the WENO schemes, address the drawbacks of the original formulation. This paper explores these improvements in the context of the CRWENO schemes and compares the different formulations of the non-linear weights for flow problems with small length scales as well as discontinuities. Simplified one- and two-dimensional inviscid flow problems are solved to demonstrate the numerical properties of the CRWENO schemes and its different formulations. Canonical turbulent flow problems—the decay of isotropic turbulence and the shock-turbulence interaction—are solved to assess the performance of the schemes for the direct numerical simulation of compressible, turbulent flows.

Application of Compact-Reconstruction Weighted Essentially Nonoscillatory Schemes to Compressible Aerodynamic Flows

Compact-reconstruction weighted essentially nonoscillatory schemes have lower dissipation and dispersion errors as well as higher spectral resolution than weighted essentially nonoscillatory schemes of the same order of convergence. Numerical experiments on benchmark inviscid flow problems have demonstrated improvements in the resolution and preservation of flow features such as vortices, discontinuities, and small-length-scale waves. This paper describes the integration of these schemes with a compressible, unsteady, Reynolds-averaged Navier–Stokes solver and demonstrates their performance for two- and three-dimensional flow problems. The schemes are validated and verified for domains discretized by curvilinear and overset grids. Several flow examples demonstrate improvements in the resolution of boundary-layer and wake-flow features for solutions obtained by the compact-reconstruction weighted essentially nonoscillatory schemes. The results presented indicate that these schemes are well suited to aerody...

Well-Balanced, Conservative Finite Difference Algorithm for Atmospheric Flows

The numerical simulation of meso-, convective-, and microscale atmospheric flows requires the solution of the Euler or the Navier–Stokes equations. Nonhydrostatic weather prediction algorithms often solve the equations in terms of derived quantities such as Exner pressure and potential temperature (and are thus not conservative) and/or as perturbations to the hydrostatically balanced equilibrium state. This paper presents a well-balanced, conservative finite difference formulation for the Euler equations with a gravitational source term, where the governing equations are solved as conservation laws for mass, momentum, and energy. Preservation of the hydrostatic balance to machine precision by the discretized equations is essential because atmospheric phenomena are often small perturbations to this balance. The proposed algorithm uses the weighted essentially nonoscillatory and compact-reconstruction weighted essentially nonoscillatory schemes for spatial discretization that yields high-order accurate solu...

EFFICIENT IMPLEMENTATION OF NONLINEAR COMPACT SCHEMES ON MASSIVELY PARALLEL PLATFORMS

Weighted nonlinear compact schemes are ideal for simulating compressible, turbulent flows because of their nonoscillatory nature and high spectral resolution. However, they require the solution to banded systems of equations at each time-integration step or stage. We focus on tridiagonal compact schemes in this paper. We propose an efficient implementation of such schemes on massively parallel computing platforms through an iterative substructuring algorithm to solve the tridiagonal system of equations. The key features of our implementation are that it does not introduce any parallelization-based approximations or errors and it involves minimal neighbor-to-neighbor communications. We demonstrate the performance and scalability of our approach on the IBM Blue Gene/Q platform and show that the compact schemes are efficient and have performance comparable to that of standard noncompact finite-difference methods on large numbers of processors (

High-Order Accurate Incompressible Navier-Stokes Algorithm for Vortex-Ring Interactions with Solid Wall

\sim500,000

Semi-implicit Time Integration of Atmospheric Flows with Characteristic-Based Flux Partitioning

) and small subdomain sizes (four points per dimensi...

Well-Balanced Formulation of Gravitational Source Terms for Conservative Finite-Difference Atmospheric Flow Solvers

The interaction of vortical structures with a solid wall is studied numerically in this paper. The incompressible Navier–Stokes equations in their primitive variable formulation are solved on a staggered Cartesian grid using the fractional-step algorithm. The convective terms are marched in time using the Adams–Bashforth or Runge–Kutta schemes,while the viscous terms are treated implicitly using the trapezoidal scheme.Anupwind reconstruction of the convective fluxes in their conservative form is proposed and compared against the nonconservative flux formulation, as differences are expected due to nonzero velocity divergence in the numerical solution. The performance of the algorithm is assessed on the convection of an isolated vortex. The algorithm is verified on benchmark incompressible flows and flows involving the convection, mutual interaction, and wall impingement of vortical structures. An attempt is made to simulate the impingement of two and multiple vortex rings against a solid wall to understand the effect of mutual interactions on the interactions with a solid wall. The effects of Reynolds number and initial separation between the rings are studied. These parameters are observed to affect the behavior of the vortical structures, thus affecting the formation and ejection of secondary vortices.

Compact-Reconstruction Weighted Essentially Non-Oscillatory Schemes for the Unsteady Navier-Stokes Equations

This paper presents a characteristic-based flux partitioning for the semi-implicit time integration of atmospheric flows. Nonhydrostatic models require the solution of the compressible Euler equations. The acoustic time scale is significantly faster than the advective scale, yet it is typically not relevant to atmospheric and weather phenomena. The acoustic and advective components of the hyperbolic flux are separated in the characteristic space. High-order, conservative additive Runge--Kutta methods are applied to the partitioned equations so that the acoustic component is integrated in time implicitly with an unconditionally stable method, while the advective component is integrated explicitly. The time step of the overall algorithm is thus determined by the advective scale. Benchmark flow problems are used to demonstrate the accuracy, stability, and convergence of the proposed algorithm. The computational cost of the partitioned semi-implicit approach is compared with that of explicit time integration.

Numerical Simulation of Vortex Ring Interactions with Solid Wall

Numerical simulation of atmospheric flows requires high-resolution, nonoscillatory algorithms to accurately capture all length scales. In this paper, a conservative finite-difference algorithm is proposed that uses the weighted essentially nonoscillatory and compact-reconstruction weighted essentially nonoscillatory schemes for spatial discretization. These schemes use solution-dependent interpolation stencils to yield high-order accurate nonoscillatory solutions to hyperbolic conservation laws. The Euler equations in their fundamental form (conservation of mass, momentum, and energy) are solved, thus avoiding approximations and simplifications. A well-balanced formulation of the finite-difference algorithm is proposed that preserves hydrostatically balanced equilibria to round-off errors. The algorithm is verified for benchmark atmospheric flow problems.