Mark Huitt Carpenter

• Ph.D. Combustion, Mech E. CMU
• National Aeronautics and Space Administration

A stencil-adaptive SBP-SAT finite difference scheme is shown to display su-perconvergent behavior. As proof of concept, applied to the linear advec-tion equation, it has a convergence rate O(∆x 4) in contrast to a conventional scheme, which converges at a rate O(∆x 3).

Provably stable flux reconstruction (FR) schemes are derived for partial differential equations cast in curvilinear coordinates. Specifically, energy stable flux reconstruction (ESFR) schemes are considered as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, the curvili...

Provably stable flux reconstruction (FR) schemes are derived for partial differential equations cast in curvilinear coordinates. Specifically, energy stable flux reconstruction (ESFR) schemes are considered as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, split forms...

We introduce solution dependent finite difference stencils whose coefficients adapt to the current numerical solution by minimizing the truncation error in the least squares sense. The resulting scheme has the resolution capacity of dispersion relation preserving difference stencils in under-resolved domains, together with the high order convergenc...

The entropy-conservative/stable, curvilinear, nonconforming, p-refinement algorithm for hyperbolic conservation laws of Del Rey Fernández et al. (2019) is extended from the compressible Euler equations to the compressible Navier–Stokes equations. A simple and flexible coupling procedure with planar interpolation operators between adjoining nonconfo...

In this paper, we extend the entropy conservative/stable algorithms presented by Del Rey Fernández et al. (2019) for the compressible Euler and Navier–Stokes equations on nonconforming p-refined/coarsened curvilinear grids to h/p refinement/coarsening. The main difficulty in developing nonconforming algorithms is the construction of appropriate cou...

In this paper, the entropy conservative/stable algorithms presented by Del Rey Fernandez and coauthors [18,16,17] for the compressible Euler and Navier-Stokes equations on nonconforming p-refined/coarsened curvilinear grids is extended to h/p refinement/coarsening. The main difficulty in developing nonconforming algorithms is the construction of ap...

The entropy conservative/stable algorithm of Friedrich~\etal (2018) for hyperbolic conservation laws on nonconforming p-refined/coarsened Cartesian grids, is extended to curvilinear grids for the compressible Euler equations. The primary focus is on constructing appropriate coupling procedures across the curvilinear nonconforming interfaces. A simp...

The entropy conservative, curvilinear, nonconforming, p-refinement algorithm for hyperbolic conservation laws of Del Rey Fernandez et al. (2019), is extended from the compressible Euler equations to the compressible Navier-Stokes equations. A simple and flexible coupling procedure with planar interpolation operators between adjoining nonconforming...

Methodologies are presented that enable the construction of provably linearly stable and conservative high-order discretizations of partial differential equations in curvilinear coordinates based on generalized summation-by-parts operators, including operators with dense-norm matrices. Specifically, three approaches are presented for the constructi...

We present a novel technique for the imposition of non-linear entropy conservative and entropy stable solid wall boundary conditions for the compressible Navier–Stokes equations in the presence of an adiabatic wall, or a wall with a prescribed heat entropy flow. The procedure relies on the formalism and mimetic properties of diagonal-norm, summatio...

New entropy stable spectral collocation schemes of arbitrary order of accuracy are developed for the unsteady 3-D Euler and Navier-Stokes equations on dynamic unstructured grids. To take into account the grid motion and deformation, we use an arbitrary Lagrangian-Eulerian formulation. As a result, moving and deforming hexahedral grid elements are i...

This work examines the development of an entropy conservative (for smooth solutions) or entropy stable (for discontinuous solutions) space–time discontinuous Galerkin (DG) method for systems of nonlinear hyperbolic conservation laws. The resulting numerical scheme is fully discrete and provides a bound on the mathematical entropy at any time accord...

The entropy conservative/stable staggered grid tensor-product algorithm of Parsani et al. [1]is extended to multidimensional SBP discretizations. The required SBP preserving interpolation operators are proven to exist under mild restrictions and the resulting algorithm is proven to be entropy conservative/stable as well as elementwise conservative....

The construction of high order entropy stable collocation schemes on quadrilateral and hexahedral elements has relied on the use of Gauss--Legendre--Lobatto collocation points and their equivalence with summation-by-parts (SBP) finite difference operators. In this work, we show how to efficiently generalize the construction of semidiscretely entrop...

We present a novel technique for the imposition of non-linear entropy conservative and entropy stable solid wall boundary conditions for the compressible Navier-Stokes equations in the presence of an adiabatic wall, or a wall with a prescribed heat entropy flow. The procedure relies on the formalism and on the mimetic properties of diagonal norm, s...

New entropy stable spectral collocations schemes of arbitrary order of accuracy are developed for the unsteady 3-D Euler and Navier-Stokes equations on dynamic unstructured grids. To take into account the grid motion and deformation, we use an arbitrary Lagrangian-Eulerian (ALE) formulation. As a result, moving and deforming hexahedral grid element...

This work presents an entropy stable discontinuous Galerkin (DG) spectral element approximation for systems of non-linear conservation laws with general geometric (h) and polynomial order (p) non-conforming rectangular meshes. The crux of the proofs presented is that the nodal DG method is constructed with the collocated Legendre-Gauss-Lobatto node...

The construction of high order entropy stable collocation schemes on quadrilateral and hexahedral elements has relied on the use of Gauss-Legendre-Lobatto collocation points and their equivalence with summation-by-parts (SBP) finite difference operators. In this work, we show how to efficiently generalize the construction of semi-discretely entropy...

This work examines the development of an entropy conservative (for smooth solutions) or entropy stable (for discontinuous solutions) space-time discontinuous Galerkin (DG) method for systems of non-linear hyperbolic conservation laws. The resulting numerical scheme is fully discrete and provides a bound on the mathematical entropy at any time accor...

We consider the numerical simulation of the acoustic wave equations arising from seismic applications, for which staggered grid finite difference methods are popular choices due to their simplicity and efficiency. We relax the uniform grid restriction on finite difference methods and allow the grids to be block-wise uniform with nonconforming inter...

We consider the numerical simulation of the acoustic wave equations arising from seismic applications, for which staggered grid finite difference methods are popular choices due to their simplicity and efficiency. We relax the uniform grid restriction on finite difference methods and allow the grids to be block-wise uniform with nonconforming inter...

This work presents an entropy stable discontinuous Galerkin (DG) spectral element approximation for systems of non-linear conservation laws with general geometric (h) and polynomial order (p) non-conforming rectangular meshes. The crux of the proofs presented is that the nodal DG method is constructed with the collocated Legendre-Gauss-Lobatto node...

We present an entropy stable numerical scheme subject to no-slip wall boundary conditions. To enforce entropy stability only the no-penetration boundary condition and a temperature condition is needed at a wall, and leads to an L 2 bound on the conservative variables. In this article, we take the next step and design a finite difference scheme that...

A systematic approach based on a diagonal-norm summation-by-parts (SBP) framework is presented for implementing entropy stable (SS) formulations of any order for the compressible Navier–Stokes equations (NSE). These SS formulations discretely conserve mass, momentum, energy and satisfy a mathematical entropy equality for smooth problems. They are a...

High-order numerical methods that satisfy a discrete analog of the entropy inequality are uncommon. Indeed, no proofs of nonlinear entropy stability currently exist for high-order weighted essentially nonoscillatory (WENO) finite volume or weak-form finite element methods. Herein, a new family of fourth-order WENO spectral collocation schemes is de...

A review of diagonally implicit Runge-Kutta (DIRK) methods applied to first-order ordinary differential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRK-type methods are revi...

Nonlinearly stable finite element methods of arbitrary type and order, are currently unavailable for dis-cretizations of the compressible Navier-Stokes equations. Summation-by-parts (SBP) entropy stability analysis provides a means of constructing nonlinearly stable discrete operators of arbitrary order, but is currently limited to simple element t...

Staggered grid, entropy stable discontinuous spectral collocation operators of any order are developed for the compressible Euler and Navier-Stokes equations on unstructured hexahedral elements. This generalization of previous entropy stable spectral collocation work [M. H. Carpenter, T. C. Fisher, E. J. Nielsen, and S. H. Frankel, SIAM J. Sci. Com...

Staggered grid, entropy stable discontinuous spectral collocation operators of any order are developed for Burgers' and the compressible Navier-Stokes equations on unstructured hexahedral elements. This generalization of previous entropy stable spectral collocation work, extends the applicable set of points from tensor product, Legendre-Gauss-Lobat...

Transition analysis is performed for a swept wing at a Mach number of 0.75 and chord Reynolds number of approximately 1.7x10(7), with a focus on roughness-based crossflow-transition control at high Reynolds numbers relevant to subsonic flight. The roughness-based transition control involves controlled seeding of suitable, subdominant crossflow mode...

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator....

Implementation of the Kumaresan and Tufts algorithm to liner impedance eduction in a duct with shear flow is described. The approach is based on a noncausal model of sound propagation coupled with singular value decomposition to identify the acoustic pressure modes. The performance of the algorithm is evaluated by comparing the educed impedance spe...

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the compressible Navier–Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the bo...

Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation element methods of arbitrary order for the compressible Navier-Stokes equations. The new methods are similar to strong form, nodal discontinuous Galerkin spectral elements but conserve entropy for the Euler equati...

Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation methods of arbitrary order. The new methods are closely related to discontinuous Galerkin spectral collocation methods commonly known as DGFEM, but exhibit a more general entropy stability property. Although the n...

A systematic approach is presented for developing entropy stable (SS) formulations of any order for the Navier-Stokes equations. These SS formulations discretely conserve mass, momentum, energy and satisfy a mathematical entropy inequality. They are valid for smooth as well as discontinuous flows provided sufficient dissipation is added at shocks a...

The economic and environmental benefits of laminar flow technology via reduced fuel burn of subsonic and supersonic aircraft cannot be realized without minimizing the uncertainty in drag prediction in general and transition prediction in particular. Transition research under NASA's Aeronautical Sciences Project seeks to develop a validated set of v...

The Lax-Wendroff theorem stipulates that a discretely conservative operator is necessary to accurately capture discontinuities. The discrete operator, however, need not be derived from the divergence form of the continuous equations. Indeed, conservation law equations that are split into linear combinations of the divergence and product rule form a...

Developing stable and robust high-order finite-difference schemes requires mathematical formalism and appropriate methods of analysis. In this work, nonlinear entropy stability is used to derive provably stable high-order finite-difference methods with formal boundary closures for conservation laws. Particular emphasis is placed on the entropy stab...

Since its founding, NASA has been dedicated to the advancement of aeronautics and space science. The NASA scientific and technical information (STI) program plays a key part in helping NASA maintain this important role. The NASA STI program operates under the auspices of the Agency Chief Information Officer. It collects, organizes, provides for arc...

A general strategy was presented in 2009 by Yamaleev and Carpenter (J. Comput. Phys. 228(11):4248–4272, 2009; J. Comput. Phys. 228(8):3025–3047, 2009), for constructing Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite-difference schemes on periodic domains. Fisher et al. (J. Comput. Phys. 230(10):3727–3752, 2011) provided boundary...

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 functio...

A combination of parabolized stability equations and secondary instability theory has been applied to a low-speed swept airfoil model with a chord Reynolds number of 7.15 million, with the goal of evaluating this methodology in the context of transition prediction for a known configuration for which roughness-based crossflow transition control has...

A general interface procedure is presented for multi-domain collocation methods satisfying the summation-by-parts (SBP) spatial discretization convention. Unlike more traditional operators (e.g. FEM) applied to the advection-diffusion equation, the new procedure penalizes the solution and the first p derivatives across the interface. The combined i...

Recent experience in the application of an optimized, second-order, backward-difference (BDF2OPT) tem-poral scheme is reported. The primary focus of the work is on obtaining accurate solutions of the unsteady Reynolds-averaged Navier-Stokes equations over long periods of time for aerodynamic problems of interest. The baseline flow solver under cons...

The problem of crossflow receptivity is considered in the context of a canonical 3D boundary layer (viz., the swept Hiemenz boundary layer) and a swept airfoil used recently in the SWIFT flight experiment performed at Texas A&M University. First, Hiemenz flow is used to analyze localized receptivity due to a spanwise periodic array of small amplitu...

Since its founding, NASA has been dedicated to the advancement of aeronautics and space science. The NASA scientific and technical information (STI) program plays a key part in helping NASA maintain this important role. The NASA STI Program operates under the auspices of the Agency Chief Information Officer. It collects, organizes, provides for arc...

A combination of parabolized stability equations and secondary instability theory has been applied to a low-speed swept airfoil model with a chord Reynolds number of 7.15 million, with the goals of (i) evaluating this methodology in the context of transition prediction for a known configuration for which roughness based crossflow transition control...

A third-order Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite difference scheme developed by the authors of this paper [N.K. Yamaleev, M.H. Carpenter, Third-order energy stable WENO scheme, J. Comput. Phys. 228 (2009) 3025–3047] was proven to be stable in the energy norm for both continuous and discontinuous solutions of systems...

Block-to-block interface interpolation operators are constructed for several common high-order finite difference discretizations. In contrast to conventional interpolation operators, these new interpolation operators maintain the strict stability, accuracy and conservation of the base scheme even when nonconforming grids or dissimilar operators are...

A new third-order Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite difference scheme for scalar and vector hyperbolic equations with piecewise continuous initial conditions is developed. The new scheme is proven to be linearly stable in the energy norm for both continuous and discontinuous solutions. In contrast to the existing hi...

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 Run...

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 Approximati...

A high-order accurate finite difference scheme is used to perform numerical studies on the benefit of high-order methods. The main advantage of the present technique is the possibility to prove stability for the linearized Euler equations on a multi-block domain, including the boundary conditions. The result is a robust high-order scheme for realis...

The comparison of numerical results for implicit–explicit and fully explicit Runge–Kutta time integration methods for a nozzle flow problem shows that filtering can significantly degrade the accuracy of the numerical solution for long-time integration problems. We demonstrate analytically and numerically that filtering-in-time errors become additiv...

A systematic methodology for approximating realistic three-dimensional synthetic jet actuators by using a reduced-order model based on the time-dependent compressible quasi-one-dimensional Euler equations is presented. The following major questions are addressed: 1) which three-dimensional actuator geometries are amenable to the quasi-one-dimension...

Research experience in constructing and applying higher order temporal schemes with error controllers for solving unsteady Reynolds-averaged Navier-Stokes equations is reported. The baseline flow solver under consideration uses a second-order backward difference scheme with a dual time stepping algorithm for advancing the flow solutions in time. Th...

In this paper iterative techniques for unsteady flow computations with implicit higher order time integration methods at large time steps are investigated. It is shown that with a minimal coding effort the standard non-linear multigrid method can be combined with a Newton–Krylov method leading to speed-ups in the order of 30%. Copyright © 2005 John...

A new reduced-order model of multidimensional synthetic jet actuators that combines the accuracy and conservation properties of full numerical simulation methods with the efficiency of simplified zero-order models is proposed. The multidimensional actuator is simulated by the solution of the time-dependent compressible quasi-one-dimensional Eider e...

Multiple high-order time-integration schemes are used to solve stiff test problems related to the Navier-Stokes (NS) equations. The primary objective is to determine whether high-order schemes can displace currently used second-order schemes on stiff NS and Reynolds averaged NS (RANS) problems, for a meaningful portion of the work-precision spectru...

In this paper the efficiency of iterative techniques for unsteady flow computations with implicit higher order time integration methods at large time steps are investi gated. Test case is flow around a circular cylinder at two Reynolds numbers: laminar flow at Re=1000 and turbulent flow at Re=10 6. It is shown that with a minimal coding effort the...

A collaborative approach to software development is described. The approach employs the agile development techniques: project retrospectives, Scrum status meetings, and elements of Extreme Programming to efficiently develop a cohesive and extensible software suite. The software product under development is a fluid dynamics simulator for performing...

A stable and high order accurate finite difference scheme is used to perform a numerical study on the benefit of high order methods. Numerical computations of a vortex-airfoil interaction governed by the Euler equations show that high order methods are required in order to capture the significant flow features for transient problems. © 2003 by the...

A concerted effort is underway at NASA Langley Research Center to create a benchmark for Computational Fluid Dynamic (CFD) codes, both unstructured and structured, against a data set for the hump model with actuation. The hump model was tested in the NASA Langley 0.3-m Transonic Cryogenic Tunnel. The CFD codes used for the analyses are the FUN2D (F...

A concerted effort is underway at NASA Langley Research Center to create a benchmark for Computational Fluid Dynamic (CFD) codes, both unstructured and structured, against a data set for the hump model with actuation. The hump model was tested in the NASA Langley 0.3-m Transonic Cryogenic Tunnel. The CFD codes used for the analyses are the FUN2D (F...

A comparison of four temporal integration techniques is presented in the context of a general purpose aerodynamics solver. The study focuses on the temporal effciency of high-order schemes, relative to the Backward Differentiation Formulae (BDF2) scheme. The high order algorithms used include the third-order BDF3 scheme, the fourth-order Modified E...

An overview of the current status of time dependent algorithms is presented. Special attention is given to algorithms used to predict fluid actuator flows, as well as other active and passive flow control devices. Capabilities for the next decade are predicted, and principal impediments to the progress of time-dependent algorithms are identified.

d papers from scientific and technical conferences, symposia, seminars, or other meetings sponsored or co-sponsored by NASA. . SPECIAL PUBLICATION. Scientific, technical, or historical information from NASA programs, projects, and missions, often concerned with subjects having substantial public interest. . TECHNICAL TRANSLATION. Englishlanguage tr...

The accuracy of two grid adaptation strategies, grid redistribution and local grid refinement, is examined by solving the 2-D Euler equations for the supersonic steady flow around a cylinder. Second- and fourth-order linear finite difference shock-capturing schemes, based on the Lax–Friedrichs flux splitting, are used to discretize the governing eq...

Opportunities for breakthroughs in the large-scale computational simulation and design of aerospace vehicles are presented. Computational fluid dynamics tools to be used within multidisciplinary analysis and design methods are emphasized. The opportunities stem from speedups and robustness improvements in the underlying unit operations associated w...

Opportunities for breakthroughs in the large-scale computational simulation and design of aerospace vehicles are presented. Computational fluid dynamics tools to be used within multidisciplinary analysis and design methods are emphasized. The opportunities stem from speedups and robustness improvements in the underlying unit operations associated w...

The accuracy and efficiency of several lower and higher order time integration schemes are investigated for engineering solution of the discretized unsteady compressible Navier–Stokes equations. Fully implicit methods tested are either the backward differentiation formulas (BDF) or stage-order two, explicit, singly diagonally implicit Runge–Kutta (...

In this paper we discuss the issue of conservation and convergence to weak solutions of several global schemes, including the commonly used compact schemes and spectral collocation schemes, for solving hyperbolic conservation laws. It is shown that suchschemes, if convergent boundedly almost everywhere, will converge to weak solutions. The results...

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...

An overview of the current status of time dependent algorithms is presented. Special attention is given to algorithms used to predict fluid actuator flows, as well as other active and passive flow control devices. Capabilities for the next decade are predicted, and principal impediments to the progress of time-dependent algorithms are identified.

In this paper we discuss the issue of conservation and convergence to weak solutions of several global schemes, including the commonly used compact schemes and spectral collocation schemes, for solving hyperbolic conservation laws. It is shown that such schemes, if convergent boundedly almost everywhere, will converge to weak solutions. The results...

Boundary and interface conditions are derived for high-order finite difference methods applied to multidimensional linear problems in curvilinear coordinates. Difficulties presented by the combination of multiple dimensions and varying coefficients are analyzed. In particular, problems related to nondiagonal norms, a varying Jacobian, and varying a...

The e#ciency and accuracy of several time integration schemes are investigated for the unsteady Navier-Stokes equations. This study focuses on the e#ciency of higher-order Runge-Kutta schemes in comparison with the popular Backward Di#erencing Formulations. For this comparison an unsteady two-dimensional laminar #ow problem is chosen, i.e. #ow arou...

A higher-order-accurate numerical procedure has been developed for solving incompressible Navier-Stokes equations for fluid flow problems. It is based on low-storage Runge-Kutta schemes for temporal discretization adn fourth- and sixth-order compact finite-difference schemes for spinal discretization. New insights are presented on the elimination o...

Procedures are developed for the solution of the Poisson equation in multiple domains in parallel. To guarantee stability, finite-difference schemes of arbitrarily high order which obey the summation-by-parts rule are utilized. If the Poisson equation is solved in parallel in each subdomain without a global scheme to couple all subdomains together...

Boundary and interface conditions are derived for high order finite difference methods applied to multidimensional linear problems in curvilinear coordinates. The boundary and interface conditions lead to conservative schemes and strict and strong stability provided that certain metric conditions are met.

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 contro...

Boundary and interface conditions for high-order finite difference methods applied to the constant coefficient Euler and Navier–Stokes equations are derived. The boundary conditions lead to strict and strong stability. The interface conditions are stable and conservative even if the finite difference operators and mesh sizes vary from domain to dom...

. Boundary and interface conditions for high order finite di#erence methods applied to the constant coe#cient Euler and Navier-Stokes equations are derived. The boundary conditions lead to strict and strong stability. The interface conditions are stable and conservative even if the finite di#erence operators and mesh sizes vary from domain to domai...

Stable and accurate interface conditions based on the SAT penalty method are derived for the linear advection–diffusion equation. The conditions are functionally independent of the spatial order of accuracy and rely only on the form of the discrete operator. We focus on high-order finite-difference operators that satisfy the summation-by-parts (SBP...

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 sol...