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A General Method for Solving Systems of Non-Linear Equations

The method of steepest descent is modified so that accelerated convergence is achieved near a root. It is assumed that the function of interest can be approximated near a root by a quadratic form. An eigenvector of the quadratic form is found by evaluating the function and its gradient at an arbitrary point and another suitably selected point. The terminal point of the eigenvector is chosen to lie on the line segment joining the two points. The terminal point found lies on an axis of the quadratic form. The selection of a suitable step size at this point leads directly to the root in the direction of steepest descent in a single step. Newton's root finding method not infrequently diverges if the starting point is far from the root. However, the current method in these regions merely reverts to the method of steepest descent with an adaptive step size. The current method's performance should match that of the Levenberg-Marquardt root finding method since they both share the ability to converge from a starting point far from the root and both exhibit quadratic convergence near a root. The Levenberg-Marquardt method requires storage for coefficients of linear equations. The current method which does not require the solution of linear equations requires more time for additional function and gradient evaluations. The classic trade off of time for space separates the two methods.

Nachtsheim, Philip R.

Meshless Local Petrov-Galerkin Method for Bending Problems

Recent literature shows extensive research work on meshless or element-free methods as alternatives to the versatile Finite Element Method. One such meshless method is the Meshless Local Petrov-Galerkin (MLPG) method. In this report, the method is developed for bending of beams - C1 problems. A generalized moving least squares (GMLS) interpolation is used to construct the trial functions, and spline and power weight functions are used as the test functions. The method is applied to problems for which exact solutions are available to evaluate its effectiveness. The accuracy of the method is demonstrated for problems with load discontinuities and continuous beam problems. A Petrov-Galerkin implementation of the method is shown to greatly reduce computational time and effort and is thus preferable over the previously developed Galerkin approach. The MLPG method for beam problems yields very accurate deflections and slopes and continuous moment and shear forces without the need for elaborate post-processing techniques.

Phillips, Dawn R.

On the Formulation of Weakly Singular Displacement/Traction Integral Equations; and Their Solution by the MLPG Method

In this paper, a very simple method is used to derive the weakly singular traction boundary integral equation based on the integral relationships for displacement gradients. The concept of the MLPG method is employed to solve the integral equations, especially those arising in solid mechanics. A moving Least Squares (MLS) interpolation is selected to approximate the trial functions in this paper. Five boundary integral Solution methods are introduced: direct solution method; displacement boundary-value problem; traction boundary-value problem; mixed boundary-value problem; and boundary variational principle. Based on the local weak form of the BIE, four different nodal-based local test functions are selected, leading to four different MLPG methods for each BIE solution method. These methods combine the advantages of the MLPG method and the boundary element method.

Atluri, Satya N.

Three-Dimensional High-Order Spectral Finite Volume Method for Unstructured Grids

Many areas require a very high-order accurate numerical solution of conservation laws for complex shapes. This paper deals with the extension to three dimensions of the Spectral Finite Volume (SV) method for unstructured grids, which was developed to solve such problems. We first summarize the limitations of traditional methods such as finite-difference, and finite-volume for both structured and unstructured grids. We then describe the basic formulation of the spectral finite volume method. What distinguishes the SV method from conventional high-order finite-volume methods for unstructured triangular or tetrahedral grids is the data reconstruction. Instead of using a large stencil of neighboring cells to perform a high-order reconstruction, the stencil is constructed by partitioning each grid cell, called a spectral volume (SV), into 'structured' sub-cells, called control volumes (CVs). One can show that if all the SV cells are partitioned into polygonal or polyhedral CV sub-cells in a geometrically similar manner, the reconstructions for all the SVs become universal, irrespective of their shapes, sizes, orientations, or locations. It follows that the reconstruction is reduced to a weighted sum of unknowns involving just a few simple adds and multiplies, and those weights are universal and can be pre-determined once for all. The method is thus very efficient, accurate, and yet geometrically flexible. The most critical part of the SV method is the partitioning of the SV into CVs. In this paper we present the partitioning of a tetrahedral SV into polyhedral CVs with one free parameter for polynomial reconstructions up to degree of precision five. (Note that the order of accuracy of the method is one order higher than the reconstruction degree of precision.) The free parameter will be determined by minimizing the Lebesgue constant of the reconstruction matrix or similar criteria to obtain optimized partitions. The details of an efficient, parallelizable code to solve three-dimensional problems for any order of accuracy are then presented. Important aspects of the data structure are discussed. Comparisons with the Discontinuous Galerkin (DG) method are made. Numerical examples for wave propagation problems are presented.

Liu, Yen

Deployment Simulation Methods for Ultra-Lightweight Inflatable Structures

Two dynamic inflation simulation methods are employed for modeling the deployment of folded thin-membrane tubes. The simulations are necessary because ground tests include gravity effects and may poorly represent deployment in space. The two simulation methods are referred to as the Control Volume (CV) method and the Arbitrary Lagrangian Eulerian (ALE) method. They are available in the LS-DYNA nonlinear dynamic finite element code. Both methods are suitable for modeling the interactions between the inflation gas and the thin-membrane tube structures. The CV method only considers the pressure induced by the inflation gas in the simulation, while the ALE method models the actual flow of the inflation gas. Thus, the transient fluid properties at any location within the tube can be predicted by the ALE method. Deployment simulations of three packaged tube models; namely coiled, Z-folded, and telescopically-folded configurations, are performed. Results predicted by both methods for the telescopically-folded configuration are correlated and computational efficiency issues are discussed.

Wang, John T.

Single-Transducer, Ultrasonic Imaging Method for High-Temperature Structural Materials Eliminates the Effect of Thickness Variation in the Image

NASA Lewis Research Center's Life Prediction Branch, in partnership with Sonix, Inc., and Cleveland State University, recently advanced the development of, refined, and commercialized an advanced nondestructive evaluation (NDE) inspection method entitled the Single Transducer Thickness-Independent Ultrasonic Imaging Method. Selected by R&D Magazine as one of the 100 most technologically significant new products of 1996, the method uses a single transducer to eliminate the superimposing effects of thickness variation in the ultrasonic images of materials. As a result, any variation seen in the image is due solely to microstructural variation. This nondestructive method precisely and accurately characterizes material gradients (pore fraction, density, or chemical) that affect the uniformity of a material's physical performance (mechanical, thermal, or electrical). Advantages of the method over conventional ultrasonic imaging include (1) elimination of machining costs (for precision thickness control) during the quality control stages of material processing and development and (2) elimination of labor costs and subjectivity involved in further image processing and image interpretation. At NASA Lewis, the method has been used primarily for accurate inspections of high temperature structural materials including monolithic ceramics, metal matrix composites, and polymer matrix composites. Data were published this year for platelike samples, and current research is focusing on applying the method to tubular components. The initial publicity regarding the development of the method generated 150 requests for further information from a wide variety of institutions and individuals including the Federal Bureau of Investigation (FBI), Lockheed Martin Corporation, Rockwell International, Hewlett Packard Company, and Procter & Gamble Company. In addition, NASA has been solicited by the 3M Company and Allison Abrasives to use this method to inspect composite materials that are manufactured by these companies.

Roth, Don J.

Development of a Hybrid RANS/LES Method for Turbulent Mixing Layers

Significant research has been underway for several years in NASA Glenn Research Center's nozzle branch to develop advanced computational methods for simulating turbulent flows in exhaust nozzles. The primary efforts of this research have concentrated on improving our ability to calculate the turbulent mixing layers that dominate flows both in the exhaust systems of modern-day aircraft and in those of hypersonic vehicles under development. As part of these efforts, a hybrid numerical method was recently developed to simulate such turbulent mixing layers. The method developed here is intended for configurations in which a dominant structural feature provides an unsteady mechanism to drive the turbulent development in the mixing layer. Interest in Large Eddy Simulation (LES) methods have increased in recent years, but applying an LES method to calculate the wide range of turbulent scales from small eddies in the wall-bounded regions to large eddies in the mixing region is not yet possible with current computers. As a result, the hybrid method developed here uses a Reynolds-averaged Navier-Stokes (RANS) procedure to calculate wall-bounded regions entering a mixing section and uses a LES procedure to calculate the mixing-dominated regions. A numerical technique was developed to enable the use of the hybrid RANS-LES method on stretched, non-Cartesian grids. With this technique, closure for the RANS equations is obtained by using the Cebeci-Smith algebraic turbulence model in conjunction with the wall-function approach of Ota and Goldberg. The LES equations are closed using the Smagorinsky subgrid scale model. Although the function of the Cebeci-Smith model to replace all of the turbulent stresses is quite different from that of the Smagorinsky subgrid model, which only replaces the small subgrid turbulent stresses, both are eddy viscosity models and both are derived at least in part from mixing-length theory. The similar formulation of these two models enables the RANS and LES equations to be solved with a single solution scheme and computational grid. The hybrid RANS-LES method has been applied to a benchmark compressible mixing layer experiment in which two isolated supersonic streams, separated by a splitter plate, provide the flows to a constant-area mixing section. Although the configuration is largely two dimensional in nature, three-dimensional calculations were found to be necessary to enable disturbances to develop in three spatial directions and to transition to turbulence. The flow in the initial part of the mixing section consists of a periodic vortex shedding downstream of the splitter plate trailing edge. This organized vortex shedding then rapidly transitions to a turbulent structure, which is very similar to the flow development observed in the experiments. Although the qualitative nature of the large-scale turbulent development in the entire mixing section is captured well by the LES part of the current hybrid method, further efforts are planned to directly calculate a greater portion of the turbulence spectrum and to limit the subgrid scale modeling to only the very small scales. This will be accomplished by the use of higher accuracy solution schemes and more powerful computers, measured both in speed and memory capabilities.

Georgiadis, Nicholas J.

Object Recognition using Feature- and Color-Based Methods

An improved adaptive method of processing image data in an artificial neural network has been developed to enable automated, real-time recognition of possibly moving objects under changing (including suddenly changing) conditions of illumination and perspective. The method involves a combination of two prior object-recognition methods one based on adaptive detection of shape features and one based on adaptive color segmentation to enable recognition in situations in which either prior method by itself may be inadequate. The chosen prior feature-based method is known as adaptive principal-component analysis (APCA); the chosen prior color-based method is known as adaptive color segmentation (ACOSE). These methods are made to interact with each other in a closed-loop system to obtain an optimal solution of the object-recognition problem in a dynamic environment. One of the results of the interaction is to increase, beyond what would otherwise be possible, the accuracy of the determination of a region of interest (containing an object that one seeks to recognize) within an image. Another result is to provide a minimized adaptive step that can be used to update the results obtained by the two component methods when changes of color and apparent shape occur. The net effect is to enable the neural network to update its recognition output and improve its recognition capability via an adaptive learning sequence. In principle, the improved method could readily be implemented in integrated circuitry to make a compact, low-power, real-time object-recognition system. It has been proposed to demonstrate the feasibility of such a system by integrating a 256-by-256 active-pixel sensor with APCA, ACOSE, and neural processing circuitry on a single chip. It has been estimated that such a system on a chip would have a volume no larger than a few cubic centimeters, could operate at a rate as high as 1,000 frames per second, and would consume in the order of milliwatts of power.

Duong, Tuan

Comparison of Iterative and Non-Iterative Strain-Gage Balance Load Calculation Methods

The accuracy of iterative and non-iterative strain-gage balance load calculation methods was compared using data from the calibration of a force balance. Two iterative and one non-iterative method were investigated. In addition, transformations were applied to balance loads in order to process the calibration data in both direct read and force balance format. NASA's regression model optimization tool BALFIT was used to generate optimized regression models of the calibration data for each of the three load calculation methods. This approach made sure that the selected regression models met strict statistical quality requirements. The comparison of the standard deviation of the load residuals showed that the first iterative method may be applied to data in both the direct read and force balance format. The second iterative method, on the other hand, implicitly assumes that the primary gage sensitivities of all balance gages exist. Therefore, the second iterative method only works if the given balance data is processed in force balance format. The calibration data set was also processed using the non-iterative method. Standard deviations of the load residuals for the three load calculation methods were compared. Overall, the standard deviations show very good agreement. The load prediction accuracies of the three methods appear to be compatible as long as regression models used to analyze the calibration data meet strict statistical quality requirements. Recent improvements of the regression model optimization tool BALFIT are also discussed in the paper.

Ulbrich, N.

Comparison of Traditional Design Nonlinear Programming Optimization and Stochastic Methods for Structural Design

Structural design generated by traditional method, optimization method and the stochastic design concept are compared. In the traditional method, the constraints are manipulated to obtain the design and weight is back calculated. In design optimization, the weight of a structure becomes the merit function with constraints imposed on failure modes and an optimization algorithm is used to generate the solution. Stochastic design concept accounts for uncertainties in loads, material properties, and other parameters and solution is obtained by solving a design optimization problem for a specified reliability. Acceptable solutions were produced by all the three methods. The variation in the weight calculated by the methods was modest. Some variation was noticed in designs calculated by the methods. The variation may be attributed to structural indeterminacy. It is prudent to develop design by all three methods prior to its fabrication. The traditional design method can be improved when the simplified sensitivities of the behavior constraint is used. Such sensitivity can reduce design calculations and may have a potential to unify the traditional and optimization methods. Weight versus reliabilitytraced out an inverted-S-shaped graph. The center of the graph corresponded to mean valued design. A heavy design with weight approaching infinity could be produced for a near-zero rate of failure. Weight can be reduced to a small value for a most failure-prone design. Probabilistic modeling of load and material properties remained a challenge.

Patnaik, Surya N.

Collocation and Galerkin Time-Stepping Methods

We study the numerical solutions of ordinary differential equations by one-step methods where the solution at tn is known and that at t(sub n+1) is to be calculated. The approaches employed are collocation, continuous Galerkin (CG) and discontinuous Galerkin (DG). Relations among these three approaches are established. A quadrature formula using s evaluation points is employed for the Galerkin formulations. We show that with such a quadrature, the CG method is identical to the collocation method using quadrature points as collocation points. Furthermore, if the quadrature formula is the right Radau one (including t(sub n+1)), then the DG and CG methods also become identical, and they reduce to the Radau IIA collocation method. In addition, we present a generalization of DG that yields a method identical to CG and collocation with arbitrary collocation points. Thus, the collocation, CG, and generalized DG methods are equivalent, and the latter two methods can be formulated using the differential instead of integral equation. Finally, all schemes discussed can be cast as s-stage implicit Runge-Kutta methods.

Huynh, H. T.

Flight Test Results of a GPS-Based Pitot-Static Calibration Method Using Output-Error Optimization for a Light Twin-Engine Airplane

As part of the NASA Aviation Safety Program (AvSP), a novel pitot-static calibration method was developed to allow rapid in-flight calibration for subscale aircraft while flying within confined test areas. This approach uses Global Positioning System (GPS) technology coupled with modern system identification methods that rapidly computes optimal pressure error models over a range of airspeed with defined confidence bounds. This method has been demonstrated in subscale flight tests and has shown small 2- error bounds with significant reduction in test time compared to other methods. The current research was motivated by the desire to further evaluate and develop this method for full-scale aircraft. A goal of this research was to develop an accurate calibration method that enables reductions in test equipment and flight time, thus reducing costs. The approach involved analysis of data acquisition requirements, development of efficient flight patterns, and analysis of pressure error models based on system identification methods. Flight tests were conducted at The University of Tennessee Space Institute (UTSI) utilizing an instrumented Piper Navajo research aircraft. In addition, the UTSI engineering flight simulator was used to investigate test maneuver requirements and handling qualities issues associated with this technique. This paper provides a summary of piloted simulation and flight test results that illustrates the performance and capabilities of the NASA calibration method. Discussion of maneuver requirements and data analysis methods is included as well as recommendations for piloting technique.

Martos, Borja

Continental-scale Validation of MODIS-based and LEDAPS Landsat ETM+ Atmospheric Correction Methods

The potential of Landsat data processing to provide systematic continental scale products has been demonstrated by several projects including the NASA Web-enabled Landsat Data (WELD) project. The recent free availability of Landsat data increases the need for robust and efficient atmospheric correction algorithms applicable to large volume Landsat data sets. This paper compares the accuracy of two Landsat atmospheric correction methods: a MODIS-based method and the Landsat Ecosystem Disturbance Adaptive Processing System (LEDAPS) method. Both methods are based on the 6SV radiative transfer code but have different atmospheric characterization approaches. The MODIS-based method uses the MODIS Terra derived dynamic aerosol type, aerosol optical thickness, and water vapor to atmospherically correct ETM+ acquisitions in each coincident orbit. The LEDAPS method uses aerosol characterizations derived independently from each Landsat acquisition and assumes a fixed continental aerosol type and uses ancillary water vapor. Validation results are presented comparing ETM+ atmospherically corrected data generated using these two methods with AERONET corrected ETM+ data for 95 10 km×10 km 30 m subsets, a total of nearly 8 million 30 m pixels, located across the conterminous United States. The results indicate that the MODIS-based method has better accuracy than the LEDAPS method for the ETM+ red and longer wavelength bands.

Landsat

Mending the Gap, An Effort to Aid the Transfer of Formal Methods Technology

Formal methods can be applied to many of the development and verification activities required for civil avionics software. RTCA/DO-178B, Software Considerations in Airborne Systems and Equipment Certification, gives a brief description of using formal methods as an alternate method of compliance with the objectives of that standard. Despite this, the avionics industry at large has been hesitant to adopt formal methods, with few developers have actually used formal methods for certification credit. Why is this so, given the volume of evidence of the benefits of formal methods? This presentation will explore some of the challenges to using formal methods in a certification context and describe the effort by the Formal Methods Subgroup of RTCA SC-205/EUROCAE WG-71 to develop guidance to make the use of formal methods a recognized approach.

Hayhurst, Kelly

Advances in Hyperspectral Image Classification Methods for Vegetation and Agricultural Cropland Studies

Hyperspectral data are becoming more widely available via sensors on airborne and unmanned aerial vehicle (UAV) platforms, as well as proximal platforms. While space-based hyperspectral data continue to be limited in availability, multiple spaceborne Earth-observing missions on traditional platforms are scheduled for launch, and companies are experimenting with small satellites for constellations to observe the Earth, as well as for planetary missions. Land cover mapping via classification is one of the most important applications of hyperspectral remote sensing and will increase in significance as time series of imagery are more readily available. However, while the narrow bands of hyperspectral data provide new opportunities for chemistry-based modeling and mapping, challenges remain. Hyperspectral data are high dimensional, and many bands are highly correlated or irrelevant for a given classification problem. For supervised classification methods, the quantity of training data is typically limited relative to the dimension of the input space. The resulting Hughes phenomenon, often referred to as the curse of dimensionality, increases potential for unstable parameter estimates, overfitting, and poor generalization of classifiers. This is particularly problematic for parametric approaches such as Gaussian maximum likelihood–based classifiers that have been the backbone of pixel-based multispectral classification methods. This issue has motivated investigation of alternatives, including regularization of the class covariance matrices, ensembles of weak classifiers, development of feature selection and extraction methods, adoption of nonparametric classifiers, and exploration of methods to exploit unlabeled samples via semi-supervised and active learning. Data sets are also quite large, motivating computationally efficient algorithms and implementations. This chapter provides an overview of the recent advances in classification methods for mapping vegetation using hyperspectral data. Three data sets that are used in the hyperspectral classification literature (e.g., Botswana Hyperion satellite data and AVIRIS airborne data over both Kennedy Space Center and Indian Pines) are described in Section 3.2 and used to illustrate methods described in the chapter. An additional high-resolution hyperspectral data set acquired by a SpecTIR sensor on an airborne platform over the Indian Pines area is included to exemplify the use of new deep learning approaches, and a multiplatform example of airborne hyperspectral data is provided to demonstrate transfer learning in hyperspectral image classification. Classical approaches for supervised and unsupervised feature selection and extraction are reviewed in Section 3.3. In particular, nonlinearities exhibited in hyperspectral imagery have motivated development of nonlinear feature extraction methods in manifold learning, which are outlined in Section 3.3.1.4. Spatial context is also important in classification of both natural vegetation with complex textural patterns and large agricultural fields with significant local variability within fields. Approaches to exploit spatial features at both the pixel level (e.g., co-occurrence–based texture and extended morphological attribute profiles [EMAPs]) and integration of segmentation approaches (e.g., HSeg) are discussed in this context in Section 3.3.2. Recently, classification methods that leverage nonparametric methods originating in the machine learning community have grown in popularity. An overview of both widely used and newly emerging approaches, including support vector machines (SVMs), Gaussian mixture models, and deep learning based on convolutional neural networks is provided in Section 3.4. Strategies to exploit unlabeled samples, including active learning and metric learning, which combine feature extraction and augmentation of the pool of training samples in an active learning framework, are outlined in Section 3.5. Integration of image segmentation with classification to accommodate spatial coherence typically observed in vegetation is also explored, including as an integrated active learning system. Exploitation of multisensor strategies for augmenting the pool of training samples is investigated via a transfer learning framework in Section 3.5.1.2. Finally, we look to the future, considering opportunities soon to be provided by new paradigms, as hyperspectral sensing is becoming common at multiple scales from ground-based and airborne autonomous vehicles to manned aircraft and space-based platforms.

Pasolli, Edoardo

Comparison of Two Load Prediction Methods for Strain-Gage Balances

Data from a high-capacity semi-span balance is used to perform a detailed comparison of the load prediction accuracy of two strain-gage balance load prediction methods. Both methods independently obtain their load prediction equations from multivariate least squares fits of balance calibration data. The first method is called Non-Iterative Method. This approach directly uses regression models of the individual load components of a balance for the load prediction. The second method is called Iterative Method. This alternate approach uses a load iteration equation for the load prediction that is constructed from the regression models of the gage outputs of the balance. Basic characteristics of the two methods are reviewed. Afterwards, both methods are applied to calibration and check load data of the chosen balance. Finally, selected analysis results are compared. These comparisons confirmed that the load prediction accuracy of the two methods is the same for all practical purposes.

wind tunnel test

An Overview of the Patch Integral Method (PIM), a New Heat Transfer Analysis Tool for Hypersonic Wind Tunnel Facilities at NASA Langley

NASA Langley’s hypersonic wind tunnels are heavily leveraged for planetary missions. The data collection method in these tunnels is thermography, and surface temperature measurements of the model surface are collected and reduced to produce surface heating data, as seen in Fig 1. However, during model injection, no temperature data are collected, and thus conventional, integral heat transfer methods cannot be used to solve for surface heating. A method was developed in the 1990’s to reduce this data despite the data gap, known as the step approximation method. The method assumes that the film coefficient behaves as a step function, the model is semi-infinite, and thermal properties are constant. With these simplifying assumptions, a Laplace transform can be performed to result in an equation that takes the initial temperature of the model and a temperature at some point in time to back out the film coefficient at that time. This is the method that is used in the current thermographic data reduction software, IHEAT. While computationally light-weight, the step approximation has several issues associated with it. The time-history of temperature is not accounted for, which is vital as heat transfer is an integral process. Additionally, the required semi-infinite assumption is unnecessary and might be violated during runtime. Thermal variation of material properties can have a sizeable impact on heating results and are not modeled by the method. This method also takes multiple seconds to “collapse” to a steady state, which is undesirable from both a facility and data reduction standpoint. The method is also very sensitive to the “effective time” approximation, an approximation of when heating instantaneously starts (which is a nonphysical simplification), and a small variation in this value can result in an error in heating results.

J. S. Cheatwood

A New Approach for a Wider Class of Entropy Split Methods for Compressible Gas Dynamics and MHD

The high order entropy split methods of Sjögreen & Yee [1, 2] by entropy splitting of the compressible Euler (inviscid) flux derivatives for a thermally-perfect gas are based on Harten’s entropy function [3, 4, 5]. Their derivation takes advantage of the homogeneity property of Euler flux, symmetrizable Euler flux derivatives and energy-norm stability in conjunction with high order classical spatial central, DRP (dispersion relation-preserving) [6, 7, 8] or Padé (compact) spatial discretizations [9] with summation-by-parts (SBP) operators [10]. Our entropy split methods have been proven entropy conserving and stable [1, 11, 12]. Our proofs do not rely on a two-point numerical flux, but rather only a linear difference operator is required to derive these methods. To extend the entropy split method for the MHD, we used the Godunov symmetrizable non-conservative MHD form [12, 13, 14]. These high order entropy split methods not only preserve certain physical properties of the chosen governing equations but are also known to either improve numerical stability, and/or minimize aliasing errors in long time integration of turbulent flow computations without the aid of added numerical dissipation. In our previous published work, extensive error norm comparison with grid refinement was performed to show the high accuracy performance of these methods. These studies also showed how well the entropy split methods conserve the entropy, momentum and mass, and preserve the kinetic energy for long time integration of the various flows [1, 2, 12, 13, 14]. The objective of the present work is to use a new approach to obtain a wider class of entropy split methods consisting of a two-point numerical flux portion and a non-conservative portion in such a way that the homogeneity property of the compressible Euler flux is not required. For high order classical spatial central, DRP (dispersion relation-preserving) or Padé (compact) spatial discretizations, this new approach can be proven to be entropy conservative with conservative spatial dsicretizations while at the same time allowing a wider class of symmetrizable inviscid flux derivatives. We also use this generalization to derive an entropy split scheme that is entropy conserving for the equations of MHD without the homogeneity property using the Godunov symmetrizable ideal MHD formulation [15].

High Order Physical Preserving Methods