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There is a very large number of very important situations which can be modeled with nonlinear parabolic partial differential equations (PDEs) in several dimensions. In general, these PDEs can be solved by discretizing in the spatial variables and transforming them into huge systems of ordinary differential equations (ODEs), which are very stiff. Therefore, standard explicit methods require a large number of iterations to solve stiff problems. But implicit schemes are computationally very expensive when solving huge systems of nonlinear ODEs. Several families of Extrapolated Stabilized Explicit Runge-Kutta schemes (ESERK) with different order of accuracy (3 to 6) are derived and analyzed in this work. They are explicit methods, with stability regions extended, along the negative real semi-axis, quadratically with respect to the number of stages s, hence they can be considered to solve stiff problems much faster than traditional explicit schemes. Additionally, they allow the adaptation of the step length easily with a very small cost.
Two new families of ESERK schemes (ESERK3 and ESERK6) are derived, and analyzed, in this work. Each family has more than 50 new schemes, with up to 84.000 stages in the case of ESERK6. For the first time, we also parallelized all these new variable step length and variable number of stages algorithms (ESERK3, ESERK4, ESERK5, and ESERK6). These parallelized strategies allow to decrease times significantly, as it is discussed and also shown numerically in two problems. Thus, the new codes provide very good results compared to other well-known ODE solvers. Finally, a new strategy is proposed to increase the efficiency of these schemes, and it is discussed the idea of combining ESERK families in one code, because typically, stiff problems have different zones and according to them and the requested tolerance the optimum order of convergence is different.
The Rothman–Woodroofe symmetry test statistic is revisited on the basis of independent but not necessarily identically distributed random variables. The distribution-freeness if the underlying distributions are all symmetric and continuous is obtained. The results are applied for testing symmetry in a meta-analysis random effects model. The consistency of the procedure is discussed in this situation as well. A comparison with an alternative proposal from the literature is conducted via simulations. Real data are analyzed to demonstrate how the new approach works in practice.
We discuss the testing problem of homogeneity of the marginal distributions of a continuous bivariate distribution based on a paired sample with possibly missing components (missing completely at random). Applying the well-known two-sample Crámer–von-Mises distance to the remaining data, we determine the limiting null distribution of our test statistic in this situation. It is seen that a new resampling approach is appropriate for the approximation of the unknown null distribution. We prove that the resulting test asymptotically reaches the significance level and is consistent. Properties of the test under local alternatives are pointed out as well. Simulations investigate the quality of the approximation and the power of the new approach in the finite sample case. As an illustration we apply the test to real data sets.
Domain experts regularly teach novice students how to perform a task. This often requires them to adjust their behavior to the less knowledgeable audience and, hence, to behave in a more didactic manner. Eye movement modeling examples (EMMEs) are a contemporary educational tool for displaying experts’ (natural or didactic) problem-solving behavior as well as their eye movements to learners. While research on expert-novice communication mainly focused on experts’ changes in explicit, verbal communication behavior, it is as yet unclear whether and how exactly experts adjust their nonverbal behavior. This study first investigated whether and how experts change their eye movements and mouse clicks (that are displayed in EMMEs) when they perform a task naturally versus teach a task didactically. Programming experts and novices initially debugged short computer codes in a natural manner. We first characterized experts’ natural problem-solving behavior by contrasting it with that of novices. Then, we explored the changes in experts’ behavior when being subsequently instructed to model their task solution didactically. Experts became more similar to novices on measures associated with experts’ automatized processes (i.e., shorter fixation durations, fewer transitions between code and output per click on the run button when behaving didactically). This adaptation might make it easier for novices to follow or imitate the expert behavior. In contrast, experts became less similar to novices for measures associated with more strategic behavior (i.e., code reading linearity, clicks on run button) when behaving didactically.
Elastic transmission eigenvalues and their computation via the method of fundamental solutions
(2020)
A stabilized version of the fundamental solution method to catch ill-conditioning effects is investigated with focus on the computation of complex-valued elastic interior transmission eigenvalues in two dimensions for homogeneous and isotropic media. Its algorithm can be implemented very shortly and adopts to many similar partial differential equation-based eigenproblems as long as the underlying fundamental solution function can be easily generated. We develop a corroborative approximation analysis which also implicates new basic results for transmission eigenfunctions and present some numerical examples which together prove successful feasibility of our eigenvalue recovery approach.
Innovative breeds of sugar cane yield up to 2.5 times as much organic matter as conventional breeds, resulting in a great potential for biogas production. The use of biogas production as a complementary solution to conventional and second-generation ethanol production in Brazil may increase the energy produced per hectare in the sugarcane sector. Herein, it was demonstrated that through ensiling, energy cane can be conserved for six months; the stored cane can then be fed into a continuous biogas process. This approach is necessary to achieve year-round biogas production at an industrial scale. Batch tests revealed specific biogas potentials between 400 and 600 LN/kgVS for both the ensiled and non-ensiled energy cane, and the specific biogas potential of a continuous biogas process fed with ensiled energy cane was in the same range. Peak biogas losses through ensiling of up to 27% after six months were observed. Finally, compared with second-generation ethanol production using energy cane, the results indicated that biogas production from energy cane may lead to higher energy yields per hectare, with an average energy yield of up to 162 MWh/ha. Finally, the Farm²CBG concept is introduced, showing an approach for decentralized biogas production.
In this article, a concept of implicit methods for scalar conservation laws in one or more spatial dimensions allowing also for source terms of various types is presented. This material is a significant extension of previous work of the first author (Breuß SIAM J. Numer. Anal. 43(3), 970–986 2005). Implicit notions are developed that are centered around a monotonicity criterion. We demonstrate a connection between a numerical scheme and a discrete entropy inequality, which is based on a classical approach by Crandall and Majda. Additionally, three implicit methods are investigated using the developed notions. Next, we conduct a convergence proof which is not based on a classical compactness argument. Finally, the theoretical results are confirmed by various numerical tests.
The established Hoeffding-Blum-Kiefer-Rosenblatt independence test statistic is investigated for partly not identically distributed data. Surprisingly, it turns out that the statistic has the well-known distribution-free limiting null distribution of the classical criterion under standard regularity conditions. An application is testing goodness-of-fit for the regression function in a non parametric random effects meta-regression model, where the consistency is obtained as well. Simulations investigate size and power of the approach for small and moderate sample sizes. A real data example based on clinical trials illustrates how the test can be used in applications.