A Posteriori analysis of discontinuous Galerkin schemes for systems of hyperbolic conservation laws

In this work we construct reliable a posteriori estimates for some semi- (spatially) discrete discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework, which leads to error control in the case of smooth solutions. The methodology we use is quite general and allows for a posteriori control of discontinuous Galerkin schemes with standard flux choices which appear in the approximation of conservation laws. In addition to the analysis, we conduct some numerical benchmarking to test the robustness of the resultant estimator.

On aposteriori error analysis of dG schemes approximating hyperbolic conservation laws

This contribution is concerned with aposteriori error analysis of discontinuous Galerkin (dG) schemes approximating hyperbolic conservation laws. In the scalar case the aposteriori analysis is based on the L1 contraction property and the doubling of variables technique. In the system case the appropriate stability framework is in L2, based on relative entropies. It is only applicable if one of the solutions, which are compared to each other, is Lipschitz. For dG schemes approximating hyperbolic conservation laws neither the entropy solution nor the numerical solution need to be Lipschitz. We explain how this obstacle can be overcome using a reconstruction approach which leads to an aposteriori error estimate.

Dynamical properties of S-gap shifts and other shift spaces

We study the dynamical properties of certain shift spaces. To help study these properties we introduce two new classes of shifts, namely boundedly supermultiplicative (BSM) shifts and balanced shifts. It turns out that any almost specified shift is both BSM and balanced, and any balanced shift is BSM. However, as we will demonstrate, there are examples of shifts which are BSM but not balanced. We also study the measure theoretic properties of balanced shifts. We show that a shift space admits a Gibbs state if and only if it is balanced. Restricting ourselves to S-gap shifts, we relate certain dynamical properties of an S-gap shift to combinatorial properties from expansions in non-integer bases. This identification allows us to use the machinery from expansions in non-integer bases to give straightforward constructions of S -gap shifts with certain desirable properties. We show that for any q∈(0,1) there is an S-gap shift which has the specification property and entropy q . We also use this identification to address the question, for a given q∈(0,1), how many S-gap shifts exist with entropy q? For certain exceptional values of q there is a unique S-gap shift with this entropy.

A Variance-Minimizing Filter for Large-Scale Applications

A truly variance-minimizing filter is introduced and its per for mance is demonstrated with the Korteweg– DeV ries (KdV) equation and with a multilayer quasigeostrophic model of the ocean area around South Africa. It is recalled that Kalman-like filters are not variance minimizing for nonlinear model dynamics and that four - dimensional variational data assimilation (4DV AR)-like methods relying on per fect model dynamics have dif- ficulty with providing error estimates. The new method does not have these drawbacks. In fact, it combines advantages from both methods in that it does provide error estimates while automatically having balanced states after analysis, without extra computations. It is based on ensemble or Monte Carlo integrations to simulate the probability density of the model evolution. When obser vations are available, the so-called importance resampling algorithm is applied. From Bayes’ s theorem it follows that each ensemble member receives a new weight dependent on its ‘ ‘distance’ ’ t o the obser vations. Because the weights are strongly var ying, a resampling of the ensemble is necessar y. This resampling is done such that members with high weights are duplicated according to their weights, while low-weight members are largely ignored. In passing, it is noted that data assimilation is not an inverse problem by nature, although it can be for mulated that way . Also, it is shown that the posterior variance can be larger than the prior if the usual Gaussian framework is set aside. However , i n the examples presented here, the entropy of the probability densities is decreasing. The application to the ocean area around South Africa, gover ned by strongly nonlinear dynamics, shows that the method is working satisfactorily . The strong and weak points of the method are discussed and possible improvements are proposed.

The impact of oceanic heat transport on the atmospheric circulation

A general circulation model of intermediate complexity with an idealized Earth-like aquaplanet setup is used to study the impact of changes in the oceanic heat transport on the global atmospheric circulation. Focus is on the atmospheric mean meridional circulation and global thermodynamic properties. The atmosphere counterbalances to a large extent the imposed changes in the oceanic heat transport, but, nonetheless, significant modifications to the atmospheric general circulation are found. Increasing the strength of the oceanic heat transport up to 2.5 PW leads to an increase in the global mean near-surface temperature and to a decrease in its equator-to-pole gradient. For stronger transports, the gradient is reduced further, but the global mean remains approximately constant. This is linked to a cooling and a reversal of the temperature gradient in the tropics. Additionally, a stronger oceanic heat transport leads to a decline in the intensity and a poleward shift of the maxima of both the Hadley and Ferrel cells. Changes in zonal mean diabatic heating and friction impact the properties of the Hadley cell, while the behavior of the Ferrel cell is mostly controlled by friction. The efficiency of the climate machine, the intensity of the Lorenz energy cycle and the material entropy production of the system decline with increased oceanic heat transport. This suggests that the climate system becomes less efficient and turns into a state of reduced entropy production as the enhanced oceanic transport performs a stronger large-scale mixing between geophysical fluids with different temperatures, thus reducing the available energy in the climate system and bringing it closer to a state of thermal equilibrium.

Natural disasters, economic growth and sustainable development in China: an empirical study using provincial panel data

Using a newly developed integrated indicator system with entropy weighting, we analyzed the panel data of 577 recorded disasters in 30 provinces of China from 1985–2011 to identify their links with the subsequent economic growth. Meteorological disasters promote economic growth through human capital instead of physical capital. Geological disasters did not trigger local economic growth from 1999–2011. Generally, natural disasters overall had no significant impact on economic growth from 1985–1998. Thus, human capital reinvestment should be the aim in managing recoveries, and it should be used to regenerate the local economy based on long-term sustainable development.

Effective interaction between monolayers of block copolymer compatiblizer in a polymer blend

The stability of ternary blends of two immiscible homopolymers and a block copolymer compatiblizer depends crucially on the effective interaction between the copolymermonolayers that form between the unlike homopolymer domains. Here, the interaction is calculated for blends involving A and B homopolymers of equal size with ABABdiblock copolymers of symmetric composition using both self-consistent field theory (SCFT) and strong-segregation theory (SST). If the homopolymers are larger than the copolymer molecules, an attractive interaction is predicted which would destroy the blend. This conclusion coupled with considerations regarding the elastic properties of the monolayer suggests that the optimum size of the homopolymer molecules is about 80% that of the copolymer molecule. A detailed examination of the theory demonstrates that the attraction results from the configurational entropy loss of the homopolymer molecules trapped between the copolymermonolayers. We conclude by suggesting how the monolayers can be altered in order to suppress this attraction and thus improve compatiblization.