A class of high-order Runge-Kutta-Chebyshev stability polynomials

The analytic form of a new class of factorized Runge-Kutta-Chebyshev (FRKC) stability polynomials of arbitrary order $N$ is presented. Roots of FRKC stability polynomials of degree $L=MN$ are used to construct explicit schemes comprising $L$ forward Euler stages with internal stability ensured through a sequencing algorithm which limits the internal amplification factors to $\sim L^2$. The associated stability domain scales as $M^2$ along the real axis. Marginally stable real-valued points on the interior of the stability domain are removed via a prescribed damping procedure. By construction, FRKC schemes meet all linear order conditions; for nonlinear problems at orders above 2, complex splitting or Butcher series composition methods are required. Linear order conditions of the FRKC stability polynomials are verified at orders 2, 4, and 6 in numerical experiments. Comparative studies with existing methods show the second-order unsplit FRKC2 scheme and higher order (4 and 6) split FRKCs schemes are efficient for large moderately stiff problems.Comment: 24 pages, 5 figures. Accepted for publication in Journal of Computational Physics, 22 Jul 2015. Revise

Finite depth effects on solitary waves in a floating ice sheet

A theoretical and numerical study of two-dimensional nonlinear flexural-gravity waves propagating at the surface of an ideal fluid of finite depth, covered by a thin ice sheet, is presented. The ice-sheet model is based on the special Cosserat theory of hyperelastic shells satisfying Kirchhoff׳s hypothesis, which yields a conservative and nonlinear expression for the bending force. From a Hamiltonian reformulation of the governing equations, two weakly nonlinear wave models are derived: a 5th-order Korteweg–de Vries equation in the long-wave regime and a cubic nonlinear Schrödinger equation in the modulational regime. Solitary wave solutions of these models and their stability are analysed. In particular, there is a critical depth below which the nonlinear Schrödinger equation is of focusing type and thus admits stable soliton solutions. These weakly nonlinear results are validated by comparison with direct numerical simulations of the full governing equations. It is observed numerically that small- to large-amplitude solitary waves of depression are stable. Overturning waves of depression are also found for low wave speeds and sufficiently large depth. However, solitary waves of elevation seem to be unstable in all cases

A coupled mitral valve -- left ventricle model with fluid-structure interaction

Understanding the interaction between the valves and walls of the heart is important in assessing and subsequently treating heart dysfunction. With advancements in cardiac imaging, nonlinear mechanics and computational techniques, it is now possible to explore the mechanics of valve-heart interactions using anatomically and physiologically realistic models. This study presents an integrated model of the mitral valve (MV) coupled to the left ventricle (LV), with the geometry derived from in vivo clinical magnetic resonance images. Numerical simulations using this coupled MV-LV model are developed using an immersed boundary/finite element method. The model incorporates detailed valvular features, left ventricular contraction, nonlinear soft tissue mechanics, and fluid-mediated interactions between the MV and LV wall. We use the model to simulate the cardiac function from diastole to systole, and investigate how myocardial active relaxation function affects the LV pump function. The results of the new model agree with in vivo measurements, and demonstrate that the diastolic filling pressure increases significantly with impaired myocardial active relaxation to maintain the normal cardiac output. The coupled model has the potential to advance fundamental knowledge of mechanisms underlying MV-LV interaction, and help in risk stratification and optimization of therapies for heart diseases.Comment: 25 pages, 6 figure

A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection-diffusion equations and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented

The Data Big Bang and the Expanding Digital Universe: High-Dimensional, Complex and Massive Data Sets in an Inflationary Epoch

Recent and forthcoming advances in instrumentation, and giant new surveys, are creating astronomical data sets that are not amenable to the methods of analysis familiar to astronomers. Traditional methods are often inadequate not merely because of the size in bytes of the data sets, but also because of the complexity of modern data sets. Mathematical limitations of familiar algorithms and techniques in dealing with such data sets create a critical need for new paradigms for the representation, analysis and scientific visualization (as opposed to illustrative visualization) of heterogeneous, multiresolution data across application domains. Some of the problems presented by the new data sets have been addressed by other disciplines such as applied mathematics, statistics and machine learning and have been utilized by other sciences such as space-based geosciences. Unfortunately, valuable results pertaining to these problems are mostly to be found only in publications outside of astronomy. Here we offer brief overviews of a number of concepts, techniques and developments, some "old" and some new. These are generally unknown to most of the astronomical community, but are vital to the analysis and visualization of complex datasets and images. In order for astronomers to take advantage of the richness and complexity of the new era of data, and to be able to identify, adopt, and apply new solutions, the astronomical community needs a certain degree of awareness and understanding of the new concepts. One of the goals of this paper is to help bridge the gap between applied mathematics, artificial intelligence and computer science on the one side and astronomy on the other.Comment: 24 pages, 8 Figures, 1 Table. Accepted for publication: "Advances in Astronomy, special issue "Robotic Astronomy

Non-Newtonian Rheology in Blood Circulation

Blood is a complex suspension that demonstrates several non-Newtonian rheological characteristics such as deformation-rate dependency, viscoelasticity and yield stress. In this paper we outline some issues related to the non-Newtonian effects in blood circulation system and present modeling approaches based mostly on the past work in this field.Comment: 26 pages, 5 figures, 2 table