ENO-wavelet transforms for piecewise smooth functions

We have designed an adaptive essentially nonoscillatory (ENO)-wavelet transform for approximating discontinuous functions without oscillations near the discontinuities. Our approach is to apply the main idea from ENO schemes for numerical shock capturing to standard wavelet transforms. The crucial point is that the wavelet coefficients are computed without differencing function values across jumps. However, we accomplish this in a different way than in the standard ENO schemes. Whereas in the standard ENO schemes the stencils are adaptively chosen, in the ENO-wavelet transforms we adaptively change the function and use the same uniform stencils. The ENO-wavelet transform retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies, obtaining maximum accuracy, maintained up to the discontinuities, and having a multiresolution framework and fast algorithms, all without any edge artifacts. We have obtained a rigorous approximation error bound which shows that the error in the ENO-wavelet approximation depends only on the size of the derivative of the function away from the discontinuities. We will show some numerical examples to illustrate this error estimate

Exercise-Induced Changes in Exhaled NO Differentiates Asthma With or Without Fixed Airway Obstruction From COPD With Dynamic Hyperinflation.

Asthmatic patients with fixed airway obstruction (FAO) and patients with chronic obstructive pulmonary disease (COPD) share similarities in terms of irreversible pulmonary function impairment. Exhaled nitric oxide (eNO) has been documented as a marker of airway inflammation in asthma, but not in COPD. To examine whether the basal eNO level and the change after exercise may differentiate asthmatics with FAO from COPD, 27 normal subjects, 60 stable asthmatics, and 62 stable COPD patients were studied. Asthmatics with FAO (n = 29) were defined as showing a postbronchodilator FEV(1)/forced vital capacity (FVC) ≤70% and FEV(1) less than 80% predicted after inhaled salbutamol (400 μg). COPD with dynamic hyperinflation (n = 31) was defined as a decrease in inspiratory capacity (ΔIC%) after a 6 minute walk test (6MWT). Basal levels of eNO were significantly higher in asthmatics and COPD patients compared to normal subjects. The changes in eNO after 6MWT were negatively correlated with the percent change in IC (r = −0.380, n = 29, P = 0.042) in asthmatics with FAO. Their levels of basal eNO correlated with the maximum mid-expiratory flow (MMEF % predicted) before and after 6MWT. In COPD patients with air-trapping, the percent change of eNO was positively correlated to ΔIC% (rs = 0.404, n = 31, P = 0.024). We conclude that asthma with FAO may represent residual inflammation in the airways, while dynamic hyperinflation in COPD may retain NO in the distal airspace. eNO changes after 6MWT may differentiate the subgroups of asthma or COPD patients and will help toward delivery of individualized therapy for airflow obstruction

EarthScope: Activity at Augustine Volcano

This bulletin provides information on the recent eruptive activity of Augustine Volcano in Alaska. Topics include some history of the volcano, its geologic setting as part of the Aleutian island arc, and earthquake locations as indicators of magma movement. The bulletin is also accompanied by an animated 360-degree rotation around the volcano and some background information on the EarthScope Project Educational levels: Middle school, High school

EarthScope: Earthquake in Tonga

This bulletin discusses the magnitude 7.9 earthquake that shook the islands of Tonga in May 2006. Topics include the tectonic setting of the islands, at the boundary of the Pacific and Australian plates, the frequent seismic activity at this location, how earthquakes are produced at subduction zones, and how they can reveal information about the Earth's interior. There is also information about the instruments used to study earthquakes and crustal movements, and some background information about the EarthScope Project. Educational levels: Middle school, High school

On the application of ENO scheme with subcell resolution to conservation laws with stiff source terms

Two approaches are used to extend the essentially non-oscillatory (ENO) schemes to treat conservation laws with stiff source terms. One approach is the application of the Strang time-splitting method. Here the basic ENO scheme and the Harten modification using subcell resolution (SR), ENO/SR scheme, are extended this way. The other approach is a direct method and a modification of the ENO/SR. Here the technique of ENO reconstruction with subcell resolution is used to locate the discontinuity within a cell and the time evolution is then accomplished by solving the differential equation along characteristics locally and advancing in the characteristic direction. This scheme is denoted ENO/SRCD (subcell resolution - characteristic direction). All the schemes are tested on the equation of LeVeque and Yee (NASA-TM-100075, 1988) modeling reacting flow problems. Numerical results show that these schemes handle this intriguing model problem very well, especially with ENO/SRCD which produces perfect resolution at the discontinuity

On the application of subcell resolution to conservation laws with stiff source terms

LeVeque and Yee recently investigated a one-dimensional scalar conservation law with stiff source terms modeling the reacting flow problems and discovered that for the very stiff case most of the current finite difference methods developed for non-reacting flows would produce wrong solutions when there is a propagating discontinuity. A numerical scheme, essentially nonoscillatory/subcell resolution - characteristic direction (ENO/SRCD), is proposed for solving conservation laws with stiff source terms. This scheme is a modification of Harten's ENO scheme with subcell resolution, ENO/SR. The locations of the discontinuities and the characteristic directions are essential in the design. Strang's time-splitting method is used and time evolutions are done by advancing along the characteristics. Numerical experiment using this scheme shows excellent results on the model problem of LeVeque and Yee. Comparisons of the results of ENO, ENO/SR, and ENO/SRCD are also presented

Eno-Osher schemes for Euler equations

The combination of the Osher approximate Riemann solver for the Euler equations and various ENO schemes is discussed for one-dimensional flow. The three basic approaches, viz. the ENO scheme using primitive variable reconstruction, either with Cauchy-Kowalewski procedure for time integration or the TVD Runge-Kutta scheme, and the flux-ENO method are tested on different shock tube cases. The shock tube cases were chosen to present a serious challenge to the ENO schemes in order to test their ability to capture flow discontinuities, such as shocks. Also the effect of the ordering of the eigen values, viz. natural or reversed ordering, in the Osher scheme is investigated. The ENO schemes are tested up to fifth order accuracy in space and time. The ENO-Osher scheme using the Cauchy-Kowalewski procedure for time integration is found to be the most accurate and robust compared with the other methods and is also computationally efficient. The tests showed that the ENO schemes perform reasonably well, but have problems in cases where two discontinuities are close together. In that case there are not enough points in the smooth part of the flow to create a non-oscillatory interpolation

Essentially non-oscillatory shock capturing methods applied to turbulence amplification in shock wave calculations

ENO (essentially non-oscillatory) schemes can provide uniformly high order accuracy right up to discontinuities while keeping sharp, essentially non-oscillatory shock transitions. Recently, an efficient implementation of ENO schemes was obtained based on fluxes and TVD Runge-Kutta time discretizations. The resulting code is very simple to program for multi-dimensions. ENO schemes are especially suitable for computing problems with both discontinuities and fine structures in smooth regions, such as shock interaction with turbulence, for which results for 1-D and 2-D Euler equations are presented. Much better resolution is observed by using third order ENO schemes than by using second order TVD schemes for such problems