Mark H. Carpenter

Several new numerical methods for compressible shear-layer simulations

An investigation is conducted of several numerical schemes for use in the computation of two-dimensional, spatially evolving, laminar, variable-density compressible shear layers. Schemes with various temporal accuracies and arbitrary spatial accuracy for both inviscid and viscous terms are presented and analyzed. All integration schemes make use of explicit or compact finite-difference derivative operators. Three classes of schemes are considered: an extension of MacCormacks original second-order temporally accurate method,a new third-order temporally accurate variant of the coupled space–time schemes proposed by Rusanov and by Kutler et al. (RKLW), and third- and fourth-order Runge–Kutta schemes. The RKLW scheme offers the simplicity and robustness of the MacCormack schemes and gives the stability domain and the nonlinear third-order temporal accuracy of the Runge–Kutta method. In each of the schemes, stability and formal accuracy are considered for the interior operators on the convection–diffusion equation Ut+aUx=avUxx, for which a and αv are constant. Both spatial and temporal accuracies are verified on the equation Ut=[b(x)Ux]x, as well as on Ut+Fx=0. Numerical boundary treat ments of various orders of accuracy are chosen and evaluated for asymptotic stability. Formally accurate boundary conditions are derived for explicit sixth-order, pentadiagonal sixth-order, and explicit, tridiagonal, and pentadiagonal eighth-order central-difference operators when used in conjunction with Runge–Kutta integrators. Damping of high wavenumber, nonphysical information is accomplished for all schemes with the use of explicit filters, derived up to sixth order on the boundaries and twelfth order in the interior. Several schemes are used to compute variable-density compressible shear layers, where regions of large gradients of flowfield variables arise near and away from the shear-layer centerline. Results indicate that in the present simulations, the effects of differences in temporal and spatial accuracy between the schemes were less important than the filtering effects. Extended MacCormack schemes were very robust, but were inefficient because of restrictive CFL limits. The third-order temporally accurate RKLW schemes were less dissipative, but had shorter run times. Runge–Kutta integrators did not possess sufficient dissipation to be useful candidates for the computation of variable-density compressible shear layers at the levels of resolution used in the current work.

Additive Runge-Kutta schemes for convection-diffusion-reaction equations

Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection-diffusion-reaction (CDR) equations. Accuracy, stability, conservation, and dense-output are first considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, (N = 2), additive Runge-Kutta (ARK2) methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms of the partitioned method are of equal order to those of the elemental methods. Derived ARK2 methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z[I] → -∞, and retain high stability efficiency in the absence of stiffness, z[I] → 0. Extrapolation-type stage-value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK2 error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK2 methods.

Low-storage, explicit Runge-Kutta schemes for the compressible Navier-Stokes equations

The derivation of low-storage, explicit Runge-Kutta (ERK) schemes has been performed in the context of integrating the compressible Navier-Stokes equations via direct numerical simulation. Optimization of ERK methods is done across the broad range of properties, such as stability and accuracy efficiency, linear and nonlinear stability, error control reliability, step change stability, and dissipation/dispersion accuracy, subject to varying degrees of memory economization. Following van der Houwen and Wray, 16 ERK pairs are presented using from two to five registers of memory per equation, per grid point and having accuracies from third- to fifth-order. Methods have been assessed using the differential equation testing code DETEST, and with the 1D wave equation. Two of the methods have been applied to the DNS of a compressible jet as well as methane-air and hydrogen-air flames. Derived 3(2) and 4(3) pairs are competitive with existing full-storage methods. Although a substantial efficiency penalty accompanies use of two- and three-register, fifth-order methods, the best contemporary full-storage methods can be nearly matched while still saving two to three registers of memory.

A stable high-order finite difference scheme for the compressible Navier-Stokes equations, far-field boundary conditions

We construct a stable high-order finite difference scheme for the compressible Navier-Stokes equations, that satisfy an energy estimate. The equations are discretized with high-order accurate finite difference methods that satisfy a Summation-By-Parts rule. The boundary conditions are imposed with penalty terms known as the Simultaneous Approximation Term technique. The main result is a stability proof for the full three-dimensional Navier-Stokes equations, including the boundary conditions. We show the theoretical third-, fourth-, and fifth-order convergence rate, for a viscous shock, where the analytic solution is known. We demonstrate the stability and discuss the non-reflecting properties of the outflow conditions for a vortex in free space. Furthermore, we compute the three-dimensional vortex shedding behind a circular cylinder in an oblique free stream for Mach number 0.5 and Reynolds number 500.

The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: a study of the boundary error

The conventional method of imposing time dependent boundary conditions for Runge-Kutta (RK) time advancement reduces the formal accuracy of the space-time method to first order locally, and second order globally, independently of the spatial operator. This counter intuitive result is analyzed in this paper. Two methods of eliminating this problem are proposed for the linear constant coefficient case: 1) impose the exact boundary condition only at the end of the complete RK cycle, 2) impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives. The first method, while retaining the RK accuracy in all cases , results in a scheme with much reduced CFL condition, rendering the RK scheme less attractive. The second method retains the same allowable time step as the periodic problem. However it is a general remedy only for the linear case. For non-linear hyperbolic equations the second method is effective only for for RK schemes of third order accuracy or less. Numerical studies are presented to verify the efficacy of each approach.

Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes

Despite the popularity of high-order explicit Runge-Kutta (ERK) methods for integrating semi-discrete systems of equations, ERK methods suffer from severe stability-based time step restrictions for very stiff problems. We implement a discontinuous Galerkin finite element method (DGFEM) along with recently introduced high-order implicit-explicit Runge-Kutta (IMEX-RK) schemes to overcome geometry-induced stiffness in fluid-flow problems. The IMEX algorithms solve the non-stiff portions of the domain using explicit methods, and isolate and solve the more expensive stiff portions using an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta method (ESDIRK). Furthermore, we apply adaptive time-step controllers based on the embedded temporal error predictors. We demonstrate in a number of numerical test problems that IMEX methods in conjunction with efficient preconditioning become more efficient than explicit methods for systems exhibiting high levels of grid-induced stiffness.

Accuracy of shock capturing in two spatial dimensions

An assessment of the accuracy of shock-capturing schemes is made for two-dimensional steady flow around a cylindrical projectile. Both a linear fourth-order method and a nonlinear third-order method are used in this study. This study shows, contrary to conventional wisdom, that captured two-dimensional shocks are asymptotically first order, regardless of the design accuracy of the numerical method. The practical implications of this finding are discussed in the context of the efficacy of high-order numerical methods for discontinuous flows

Spatial direct numerical simulation of high-speed boundary-layer flows part I: Algorithmic considerations and validation

A highly accurate algorithm for the direct numerical simulation (DNS) of spatially evolving high-speed boundary-layer flows is described in detail and is carefully validated. To represent the evolution of instability waves faithfully, the fully explicit scheme relies on non-dissipative high-order compact-difference and spectral collocation methods. Several physical, mathematical, and practical issues relevant to the simulation of high-speed transitional flows are discussed. In particular, careful attention is paid to the implementation of inflow, outflow, and far-field boundary conditions. Four validation cases are presented, in which comparisons are made between DNS results and results obtained from either compressible linear stability theory or from the parabolized stability equation (PSE) method, the latter of which is valid for nonparallel flows and moderately nonlinear disturbance amplitudes. The first three test cases consider the propagation of two-dimensional second-mode disturbances in Mach 4.5 flat-plate boundary-layer flows. The final test case considers the evolution of a pair of oblique second-mode disturbances in a Mach 6.8 flow along a sharp cone. The agreement between the fundamentally different PSE and DNS approaches is remarkable for the test cases presented.

Computational Considerations for the Simulation of Shock-Induced Sound

The numerical study of aeroacoustic problems places stringent demands on the choice of a computational algorithm because it requires the ability to propagate disturbances of small amplitude and short wavelength. The demands are particularly high when shock waves are involved because the chosen algorithm must also resolve discontinuities in the solution. The extent to which a high-order accurate shock-capturing method can be relied upon for aeroacoustics applications that involve the interaction of shocks with other waves has not been previously quantified. Such a study is initiated in this work. A fourth-order accurate essentially nonoscillatory (ENO) method is used to investigate the solutions of inviscid, compressible flows with shocks. The design order of accuracy is achieved in the smooth regions of a steady-state, quasi-one-dimensional test case. However, in an unsteady test case, only first-order results are obtained downstream of a sound-shock interaction. The difficulty in obtaining a globally high-order accurate solution in such a case with a shock-capturing method is demonstrated through the study of a simplified, linear model problem. Some of the difficult issues and ramifications for aeroacoustic simulations of flows with shocks that are raised by these results are discussed.

Stable and accurate boundary treatments for compact, high-order finite-difference schemes

Abstract The stability characteristics of various compact fourth- and sixth-order spatial operators sre used to assess the theory of Gustafsson, Kreiss and Sundstrom (G-K-S) for the semidiscrete initial boundary value problem (IBVP). In all cases, favorable comparisons are obtained between G-K-S theory, eigenvalue determination, and numerical simulation. The conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable (bounded, left half-plane (LHP) eigenvalues). It is shown that many of the higher-order schemes that are G-S-K stable are not asymptotically stable. A series of compact fourth- and sixth-order schemes is developed, all of which are asymptotically and G-K-S stable for the scalar case. A systematic technique is then presented for constructing stable and accurate boundary closures of various orders. The technique uses the semidescrete summation-by-parts energy norm to guarantee asymptotic and G-K-S stability of the resulting boundary closure. Various fourth-order explicit and implicit discretizations are presented, all of which satisfy the summation-by-parts energy norm.