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All sourcesNASA NTRSDOE OSTI
NASA NTRS2013

Spurious Behavior of Shock-Capturing Methods: Problems Containing Stiff Source Terms and Discontinuities

The goal of this paper is to relate numerical dissipations that are inherited in high order shock-capturing schemes with the onset of wrong propagation speed of discontinuities. For pointwise evaluation of the source term, previous studies indicated that the phenomenon of wrong propagation speed of discontinuities is connected with the smearing of the discontinuity caused by the discretization of the advection term. The smearing introduces a nonequilibrium state into the calculation. Thus as soon as a nonequilibrium value is introduced in this manner, the source term turns on and immediately restores equilibrium, while at the same time shifting the discontinuity to a cell boundary. The present study is to show that the degree of wrong propagation speed of discontinuities is highly dependent on the accuracy of the numerical method. The manner in which the smearing of discontinuities is contained by the numerical method and the overall amount of numerical dissipation being employed play major roles. Moreover, employing finite time steps and grid spacings that are below the standard Courant-Friedrich-Levy (CFL) limit on shockcapturing methods for compressible Euler and Navier-Stokes equations containing stiff reacting source terms and discontinuities reveals surprising counter-intuitive results. Unlike non-reacting flows, for stiff reactions with discontinuities, employing a time step and grid spacing that are below the CFL limit (based on the homogeneous part or non-reacting part of the governing equations) does not guarantee a correct solution of the chosen governing equations. Instead, depending on the numerical method, time step and grid spacing, the numerical simulation may lead to (a) the correct solution (within the truncation error of the scheme), (b) a divergent solution, (c) a wrong propagation speed of discontinuities solution or (d) other spurious solutions that are solutions of the discretized counterparts but are not solutions of the governing equations. The present investigation for three very different stiff system cases confirms some of the findings of Lafon & Yee (1996) and LeVeque & Yee (1990) for a model scalar PDE. The findings might shed some light on the reported difficulties in numerical combustion and problems with stiff nonlinear (homogeneous) source terms and discontinuities in general.

Yee, Helen M. C.↗
NASA NTRSDate not supplied

A Moving Discontinuous Galerkin Method with Interface Condition Enforcement for Reacting Hypersonic Flows

The necessity of enforcing conservation in computational elements or cells (element conservation) for discontinuous solutions is well understood and respected for solving conservation laws in computational fluid dynamics (CFD). In contrast, interface conservation, where the conservation across cell interfaces is enforced, is long ignored, and yet is also ruled and required by the underlying physics just like element conservation. Violation of the interface conservation across discontinuities is the root cause why an exact discontinuous solution can never be achieved in shock capturing methods. The interface conservation is examined and explored in this talk. Moving discontinuous Galerkin (MDG) finite element method with interface conservation enforcement (MDG-ICE) [1],[2] are then presented for solving compressible flow problems with discontinuities based on the observation that the interface conservation can only be satisfied, only when mesh interfaces are aligned with discontinuities. In the MDG-ICE formulation, both conservative quantities and grid geometry are considered as independent variables. A space-time DG formulation is used to solve the multi-material compressible Euler equations in the standard discontinuous solution space and the discrete grid geometry is solved using a variational formulation in a continuous space. A self-adaptive Levenberg-Marquardt method is utilized to solve the resulting over-determined system of nonlinear equations arising from the MDG-ICE formulation. A number of numerical experiments for a variety of flow problems are conducted to assess the accuracy and performance of the MDG-ICE method. Numerical results obtained indicate that the MDG-ICE method is able to deliver the designed order of both h- and p-convergence even for discontinuous solutions, and detect all types of interfaces, via interface condition enforcement and satisfy, via grid movement, the compressible Euler equations and the associated interface condition.

Discontinuous Galekin↗
NASA NTRS1979

Magnetic field directional discontinuities. 1: Minimum variance errors

Errors associated with the minimum variance analysis of directional discontinuity normal components were investigated using both computer simulation of discontinuities with controlled properties and the examination of current sheets observed by the Mariner 10 spacecraft. The simulated discontinuities were created by adding fluctuations, represented by isotropic noise, to exactly known but varying (in a plane) magnetic field components. An empirical expression for the magnitude of the error in an estimated discontinuity normal component, relative to the total field across the discontinuity, was derived, as well as other relevant statistical properties. Use of the empirical relation in the analysis of 644 discontinuities observed by Mariner 10 provides a more precise, but probably conservative, estimate of an upper bound on the relative normal component value for tangential discontinuities that can be used to separate rotational from tangential discontinuities in studies using only magnetic field data from a single spacecraft, at least for the interplanetary region of space considered.

Lepping, R. P.↗
NASA NTRS2001

Nature of Fluctuations on Directional Discontinuities Inside a Solar Ejection: Wind and IMP 8 Observations

A solar ejection passed the Wind spacecraft between December 23 and 26, 1996. On closer examination, we find a sequence of ejecta material, as identified by abnormally low proton temperatures, separated by plasmas with typical solar wind temperatures at 1 AU. Large and abrupt changes in field and plasma properties occurred near the separation boundaries of these regions. At the one boundary we examine here, a series of directional discontinuities was observed. We argue that Alfvenic fluctuations in the immediate vicinity of these discontinuities distort minimum variance normals, introducing uncertainty into the identification of the discontinuities as either rotational or tangential. Carrying out a series of tests on plasma and field data including minimum variance, velocity and magnetic field correlations, and jump conditions, we conclude that the discontinuities are tangential. Furthermore, we find waves superposed on these tangential discontinuities (TDs). The presence of discontinuities allows the existence of both surface waves and ducted body waves. Both probably form in the solar atmosphere where many transverse nonuniformities exist and where theoretically they have been expected. We add to prior speculation that waves on discontinuities may in fact be a common occurrence. In the solar wind, these waves can attain large amplitudes and low frequencies. We argue that such waves can generate dynamical changes at TDs through advection or forced reconnection. The dynamics might so extensively alter the internal structure that the discontinuity would no longer be identified as tangential. Such processes could help explain why the occurrence frequency of TDs observed throughout the solar wind falls off with increasing heliocentric distance. The presence of waves may also alter the nature of the interactions of TDs with the Earth's bow shock in so-called hot flow anomalies.

Vasquez, Bernard J.↗
NASA NTRS1973

Identification of interplanetary tangential and rotational discontinuities.

Discontinuities in the interplanetary magnetic field detected by the Mariner 5 magnetometer have been analyzed to distinguish tangential and rotational discontinuities. The method used was based on finding the field component normal to the plane of the discontinuities that vanishes for tangential discontinuities and is nonzero for rotational discontinuities. Slightly less than one half of the 118 discontinuities analyzed could not be classified because neither the normal component nor the changes in field magnitude were statistically significant owing to errors in the field measurements and in finding the normal direction. The 44 rotational discontinuities that were finally identified did not exhibit changes in field magnitude in excess of 20%.-

Smith, E. J.↗
NASA NTRS1973

Variations of the occurrence rate of discontinuities in the interplanetary magnetic field

A statistical study of magnetic field discontinuities for the interval from February to October 1968 using data from the NASA Goddard Space Flight Center/University of Rome experiment on Pioneer 8 has been conducted. High-resolution data (10-sec averages) have been employed, stringent criteria being utilized to select more than 16,000 events, about 50 per cent being identified as disturbances that are probably tangential discontinuities (TD). Near 1 AU the average occurrence rate of discontinuities is about 3.6/hr, whereas that of tangential-like discontinuities is about 1.6/hr. The rate of occurrence of discontinuities is found to be correlated with the directional change of the field across the discontinuity surface. The normal to the TD discontinuity plane is most often perpendicular to the Archimedean spiral field configuration. A decrease in the rate of occurrence of TD is found with increasing heliocentric distance and/or with decreasing heliocentric latitude. Possible interpretations of the effect are discussed.

Mariani, F.↗
NASA NTRS1971

Hydromagnetic waves and discontinuities in the solar wind.

Development of a theory of hydromagnetic waves and discontinuities which is appropriate for the solar wind. The experimental evidence for the various waves, discontinuities, and some of the instabilities which are predicted by this theory is reviewed. Nearly all of the discontinuities given by the theory are shown to exist in the solar wind. These include tangential discontinuities, forward and reverse fast and slow shocks, perpendicular shocks, and Alfven shocks. Parallel shocks and contact discontinuities have not been found. A number of special cases are considered which show the basic physical properties of hydromagnetic waves in an anisotropic, multifluid, collisionless plasma. A treatment of discontinuities is presented which most resembles those of Chao (1970) and Hudson (1970). On the basis of the experimental results reviewed it is concluded that hydromagnetic theory is applicable to the solar wind.

Burlaga, L. F.↗
NASA NTRS1973

Observed properties of interplanetary rotational discontinuities.

In a prior study, discontinuities were identified as tangential or rotational principally on the basis of the existence or absence of a magnetic field component perpendicular to the plane of the discontinuity. The observed properties of the rotational discontinuities are described in this paper. The normals to the discontinuities were subjected to a variance analysis and did not exhibit a statistically significant anisotropy. These normals appear to be randomly oriented. Since rotational discontinuities are large-amplitude Alfven waves, the transverse perturbation in the magnetic field implies a corresponding jump in the solar wind velocity that is often only slightly less than the Alfven speed. The rotational discontinuities identified are all circularly polarized and transport vorticity and angular momentum.

Smith, E. J.↗
NASA NTRS1987

Stimulated dissipation of magnetic discontinuities and the origin of solar flares

It is proposed that the principal cause of the confined solar flare is the dissipation of magnetic energy at the many small-scale pre-existing tangential discontinuities in the local bipolar magnetic field. The discontinuities are a consequence of the continuous shuffling and intermixing of the footpoints of the bipolar field by the turbulent photospheric granules. The X-ray corona within the bipolar field is presumed to be a consequence of the continuing dissipation by reconnection at these discontinuities. A flare results when static deformation and/or internal agitation of the field stimulates the onset of rapid reconnection at the many small internal discontinuities. The discontinuities are partially exhausted by the flare, so that the post-flare X-ray emission of that particular loop is substantially below the pre-flare level for a period of some hours while the discontinuities are being rejuvenated.

Parker, E. N.↗
NASA NTRS2023

Foreshock Ion Motion Across Discontinuities: Formation of Foreshock Transients

In the ion foreshock, hot flow anomalies (HFAs) and foreshock bubbles (FBs) are two types of foreshock transients that have the strongest fluctuations, which can disturb the magnetosphere-ionosphere system and increase shock acceleration efficiency. They form due to interaction between the foreshock ions and solar wind discontinuities: the direction of the foreshock ion-driven current and whether it decreases or increases the magnetic field strength behind the discontinuity determine whether the transient's formation can be promoted or suppressed. Thus, to predict the HFA and FB formation and forecast their space weather effects, it is necessary to predict the foreshock ion-driven current direction. In this study, we derive analytical equations of foreshock ion velocities within discontinuities to estimate foreshock ion-driven current direction, which provides a quantitative criterion of HFA and FB formation. To validate the criterion, we use Acceleration Reconnection Turbulence & Electrodynamics of Moon's Interaction with the Sun to observe pristine solar wind discontinuities and calculate discontinuity parameters. We use Magnetospheric Multiscale to observe the foreshock ion motion around the discontinuities and show that the data support our model. This study is another step toward a predictive model of HFA and FB formation so that we can forecast their space weather effects at Earth using solar wind observations at lunar orbit or L1.

Terry Z. Liu↗
NASA NTRS1972

Discontinuities and shock waves in the interplanetary medium and their interaction with the magnetosphere.

The discontinuous structure of the solar wind is described with emphasis on properties related to geomagnetic impulses. Some of the discontinuities are clearly hydromagnetic shocks and tangential discontinuities, and can produce a significant change in the momentum flux at the magnetosphere boundary. Such a change generates an impulse which propagates through the magnetosphere to the earth where it is observed world-wide as an impulse in magnetograms. The propagation process is not reviewed here, but the relation between the initial cause (discontinuity) and the final effect (geomagnetic impulse) is reviewed in detail. The various types of impulses are examined, and are related qualitatively to the various types of discontinuities. The magnitude of an impulse is related to the change in the momentum flux. The propagation time and the rise time depend on the propagation process rather than on the initial state.

Burlaga, L. F.↗
NASA NTRS1973

On the relation between solar wind structure and solar wind rotational and tangential discontinuities.

Solar wind plasma and magnetic field data were used to identify rotational and tangential discontinuities during the first 40 days of the flight of Mariner 5. Of the 40 rotational discontinuities found, 36 were clustered in three distinct 3- to 6-day intervals. These three intervals were characterized by high solar wind bulk velocities, high magnetic field magnitudes, low densities, high correlation between velocity and magnetic field changes, and the presence of smooth Alfven waves. The rotational discontinuities had magnetic field changes that were generally close to the direction of magnetic field changes for smooth Alfven waves. Although the primary component of these changes was perpendicular to the equatorial plane for both, the radial component for the discontinuities was generally larger. The normals to the tangential discontinuity surfaces strongly preferred a direction near the ecliptic plane and perpendicular to the spiral field direction.

Turner, J. M.↗
NASA NTRS1977

Plasma field characteristics of directional discontinuities in the interplanetary medium

The paper examines plasma and magnetic-field changes occurring across 1359 directional discontinuities taken from interplanetary data spanning almost four solar rotations. The plasma field characteristics of these events exhibit a distinct variation with large-scale solar-wind velocity. At low velocities, tangential discontinuities appear to predominate. At higher velocities, a substantial and increasing fraction of directional discontinuities exhibits the plasma field properties expected of outwardly propagating rotational discontinuities. The results of Sari (1972, 1975) and of the present study suggest that in the calculation of propagation diffusion coefficients for low-energy cosmic rays, the effects of directional discontinuities should be subtracted from the magnetic fluctuation spectrum during relatively quiet wind conditions. It is not clear that such subtraction is necessary during more disturbed periods.

Solodyna, C. V.↗
NASA NTRS1977

The artificial compression method for computation of shocks and contact discontinuities. I - Single conservation laws

The paper discusses the use of the artificial compression method for the computation of discontinuous solutions of a single conservation law by finite difference methods. The single conservation law has either a shock or a contact discontinuity. Any monotone finite difference scheme applied to the original equation smears the discontinuity, while the same scheme applied to the equation modified by an artificial compression flux produces steady progressing profiles. If L is any finite difference scheme in conservation form and C is an artificial compressor, the split flux artificial compression method CL is a corrective scheme: L smears the discontinuity while propagating it; C compresses the smeared transition toward a sharp discontinuity. Numerical implementation of artificial compression is described.

Harten, A.↗
NASA NTRS1977

Discontinuous transitions in a current-carrying plasma

The properties of 'discontinuous' transitions in a current-carrying plasma are investigated by adopting an approach based on the two-fluid equations for a plasma. Braginskii's (1965) transport equations for a two-fluid plasma carrying a current are presented, and allowance is made for collisions with a background neutral gas as well as for effects of reflected particles. The speed of sound in a current-carrying plasma is derived so that the Mach number can be defined correctly as the ratio of the flow speed to the wave phase velocity in such a plasma. The various types of discontinuities permitted by the relevant jump relations in the absence of reflected particles are classified, the effect of reflected particles on such discontinuities is examined, and it is shown that the degree of indeterminacy of a discontinuity can be changed by the presence of reflected particles. Consideration of shocks containing a population of reflected particles and adiabatic transitions indicates that the pressure of reflected particles alone can give rise to both weak and strong transitions. The construction of discontinuous periodic flow solutions is also discussed.

Lee, H.-J.↗
NASA NTRS1978

The artificial compression method for computation of shocks and contact discontinuities. III - Self-adjusting hybrid schemes

This paper presents a new computational method for the calculation of discontinuous solutions of hyperbolic systems of conservation laws, which deal effectively with both shock and contact discontinuities. The method consists of two stages: in the first stage a standard finite-difference scheme is hybridized with a nonoscillatory first order accurate method to provide for the monotonic variation of the solution near discontinuities, and in the second stage artificial compression is applied to sharpen transitions at discontinuities. This modification of a standard finite-difference method results in a scheme which preserves the order of truncation error of the original method and yet yields a sharp and oscillation free transition for both shocks and contact discontinuities. The modification can be easily implemented in existing computer codes.

Harten, A.↗
NASA NTRS1980

Magnetic field directional discontinuities. I - Minimum variance errors

The paper deals with a statistical analysis of the errors associated with a minimum variance analysis of directional discontinuities by use of an idealized model of these discontinuities and various simulations, and also by an examination of actual Mariner 10 interplanetary magnetic field data. An empirical expression is derived for the magnitude of the error in an estimated discontinuity normal component, relative to the total field across the directional discontinuity. The analysis was performed primarily to aid in differentiating between interplanetary tangential and rotational discontinuities observed by Mariner 10.

Lepping, R. P.↗
NASA NTRS1980

Magnetic field directional discontinuities. 2: Characteristics between 0.46 and 1.0 AU

The characteristics of directional discontinuities (DD's) in the interplanetary magnetic field are studied using data from the Mariner 10 primary mission between 1.0 and 0.46 AU. Statistical and visual survey methods for DD identification resulted in a total of 644 events. Two methods were used to estimate the ratio of the number of tangential discontinuities (TD's) to the number of rotational discontinuities (RD's). Both methods show that the ratio of TD's to RD's varied with time and decreased with decreasing radial distance. A decrease in average discontinuity thickness of approx. 40 percent was found between 1.0 and 0.72 AU and approx. 54 percent between 1.0 and 0.46 AU, independent of type (TD or RD). This decrease in thickness for decreasing r is in qualitative agreement with Pioneer 10 observations between 1 and 5 AU. When the individual DD thickness are normalized with respect to the estimated local proton gyroradius (RA sub L), the average thickness at the three locations is nearly constant, 43 + or - 6 R sub L. This also holds true for both RD's and TD's separately. Statistical distributions of other properties, such as normal components and discontinuity plane angles, are presented.

Lepping, R. P.↗