In this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations. Previously termed TVD (total variation diminishing) time discretizations, these high-order time discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations. The new developments in this paper include the construction of optimal explicit SSP linear Runge--Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge--Kutta and multistep methods.

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Strong Stability-Preserving High-Order Time Discretization Methods

In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in Shu& Osher (1988), suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.

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Total variation diminishing Runge-Kutta schemes

In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate finite difference schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics.

https://www3.nd.edu/~zxu2/acms60790S13/Shu-WENO-notes.pdf

Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

In recent years, the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS) has been redesigned to better enable the use of application-specific and third-party algebraic solvers and data structures. Throughout this work, we have adhered to specific guiding principles that minimized the impact to current users while providing maximum flexibility for later evolution of solvers and data structures. The redesign was done through creation of new classes for linear and nonlinear solvers, enhancements to the vector class, and the creation of modern Fortran interfaces that leverage interoperability features of the Fortran 2003 standard. The vast majority of this work has been performed "behind-the-scenes," with minimal changes to the user interface and no reduction in solver capabilities or performance. However, these changes now allow advanced users to create highly customized solvers that exploit their problem structure, enabling SUNDIALS use on extreme-scale, heterogeneous computational architectures.

Enabling New Flexibility in the SUNDIALS Suite of Nonlinear and Differential/Algebraic Equation Solvers.

Accurate details of the general performance of fluid actuators is desirable over a range of flow conditions, within some predetermined error tolerance. Designers typically model actuators with different levels of fidelity depending on the acceptable level of error in each circumstance. Crude properties of the actuator (e.g., peak mass rate and frequency) may be sufficient for some designs, while detailed information is needed for other applications (e.g., multiple actuator interactions). This work attempts to address two primary objectives. The first objective is to develop a systematic methodology for approximating realistic 3-D fluid actuators, using quasi-1-D reduced-order models. Near full fidelity can be achieved with this approach at a fraction of the cost of full simulation and only a modest increase in cost relative to most actuator models used today. The second objective, which is a direct consequence of the first, is to determine the approximate magnitude of errors committed by actuator model approximations of various fidelities. This objective attempts to identify which model (ranging from simple orifice exit boundary conditions to full numerical simulations of the actuator) is appropriate for a given error tolerance.

/pdf/a-reduced-order-model-for-zero-mass-synthetic-jet-actuators-1gwv0ciuuq.pdf

A Reduced-Order Model For Zero-Mass Synthetic Jet Actuators

A systematic approach for constructing Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite difference schemes of arbitrary order is presented. The new class of schemes is proven to be stable in the energy norm for both continuous and discontinuous solutions of systems of linear hyperbolic equations. We also present new weight functions and constraints on their parameters, which provide consistency and much faster convergence of the high-order ESWENO schemes to their underlying linear schemes. Furthermore, the improved weight functions guarantee that the ESWENO schemes are design-order accurate for smooth solutions with arbitrary number of vanishing derivatives and provide much better resolution near strong discontinuities than the conventional counterparts. Numerical results show that the new ESWENO schemes are stable and significantly outperform the corresponding WENO schemes of Jiang and Shu in terms of accuracy, while providing essentially non-oscillatory solutions near strong discontinuities.

Energy Stable WENO Schemes of Arbitrary Order

The physical phenomena involved when a supersonic flow undergoes chemical reaction are discussed. Detailed physical models of convective and diffusive mixing, and finite rate chemical reaction in supersonic flow are presented. Numerical algorithms used to solve the equations governing these processes are introduced. Computer programs using these algorithms are used to analyze the structure of the reacting mixing layer. It is concluded that, as in subsonic flow, exothermic heat release in unconfined supersonic flows retards fuel/air mixing. Non mixing is shown to be a potential problem in reducing the efficiency of supersonic as well as subsonic combustion. Techniques for enhancing fuel/air mixing and combustion are described.

Supersonic combustor modeling

The linear amplification of modal disturbances that lead to boundary-layer transition in two-dimensional/axisymmetric hypersonic configurations is strongly reduced by the presence of a blunt nosetip, and the mechanisms underlying the observed onset of transition over the cone frustum are currently unknown. Linear nonmodal analysis has shown that both planar and oblique traveling disturbances that peak within the entropy layer experience appreciable energy amplification for moderate to large nosetip bluntness. The present study extends the previous linear analysis by including the nonlinear effects. Specifically, the perturbation form of the 2D, harmonic Navier-Stokes equations (HNSE) are solved with a fully implicit formulation and the Newton-Raphson method. The increased number of degrees of freedom for the nonlinear system presents difficulties for solution strategies based on direct solution of the linearized system. Such difficulties are overcome by using the GMRES iterative method with a preconditioner corresponding to a simplified Jacobian without the cross derivative terms. The HNSE solver is verified by comparing with nonlinear parabolized stability equation (NPSE) results for the nonlinear evolution of planar waves in an incompressible Blasius boundary layer and in a Mach 6 flow over a blunt cone. Finally, nonlinear nonmodal results are presented for planar traveling disturbances over the blunt cone. The nonmodal analysis demonstrates that entropy-layer disturbances generated close to the nose tip can seed the amplification of higher frequency Mack’s second-mode instabilities further downstream. 

Mark H. Carpenter

Papers

A Reduced-Order Model For Zero-Mass Synthetic Jet Actuators

Energy Stable WENO Schemes of Arbitrary Order

Supersonic combustor modeling

Nonlinear Nonmodal Analysis of Hypersonic Flow over Blunt Cones

Accurate Schemes for the Numerical Simulation of Incompressible Flows