Solution strategies for nonlinear conservation laws

Nonlinear conservation laws form the basis for models for a wide range of physical phenomena. Finding an optimal strategy for solving these problems can be challenging, and a good strategy for one problem may fail spectacularly for others. As different problems have different challenging features, exploiting knowledge about the problem structure is a key factor in achieving an efficient solution strategy. Most strategies found in literature for solving nonlinear problems involve a linearization step, usually using Newton's method, which replaces the original nonlinear problem by an iteration process consisting of a series of linear problems. A large effort is then spent on finding a good strategy for solving these linear problems. This involves choosing suitable preconditioners and linear solvers. This approach is in many cases a good choice and a multitude of different methods have been developed. However, the linearization step to some degree involves a loss of information about the original problem. This is not necessarily critical, but in many cases the structure of the nonlinear problem can be exploited to a larger extent than what is possible when working solely on the linearized problem. This may involve knowledge about dominating physical processes and specifically on whether a process is near equilibrium. By using nonlinear preconditioning techniques developed in recent years, certain attractive features such as automatic localization of computations to parts of the problem domain with the highest degree of nonlinearities arise. In the present work, these methods are further refined to obtain a framework for nonlinear preconditioning that also takes into account equilibrium information. This framework is developed mainly in the context of porous media, but in a general manner, allowing for application to a wide range of problems. A scalability study shows that the method is scalable for challenging two-phase flow problems. It is also demonstrated for nonlinear elasticity problems. Some models arising from nonlinear conservation laws are best solved using completely different strategies than the approach outlined above. One such example can be found in the field of surface gravity waves. For special types of nonlinear waves, such as solitary waves and undular bores, the well-known Korteweg-de Vries (KdV) equation has been shown to be a suitable model. This equation has many interesting properties not typical of nonlinear equations which may be exploited in the solver, and strategies usually reserved to linear problems may be applied. In this work includes a comparative study of two discretization methods with highly different properties for this equation

Domain decomposition preconditioning for non-linear elasticity problems

We consider domain decomposition techniques for a non-linear elasticity problem. Our main focus is on non-linear preconditioning, realized in the framework of additive Schwarz preconditioned inexact Newton (ASPIN) methods. The standard 1-level ASPIN method is extended to a 2-level method by adding a non-linear coarse solver. Numerical experiments show that the coarse component is necessary for scalability in terms of linear iterations inside the Newton loop. Moreover, for problems that are dominated by nonlinearities that are not localized in space the non-linear coarse iterations are crucial for achieving computational efficiency.publishedVersio

Two-scale preconditioning for two-phase nonlinear flows in porous media

Solving realistic problems related to flow in porous media to desired accuracy may be prohibitively expensive with available computing resources. Multiscale effects and nonlinearities in the governing equations are among the most important contributors to this situation. Hence, developing methods that handle these features better is essential in order to be able to solve the problems more efficiently. Focus has until recently largely been on preconditioners for linearized problems. This article proposes a two-scale nonlinear preconditioning technique for flow problems in porous media that allows for incorporating physical intuition directly in the preconditioner. By assuming a certain dominant physical process, this technique will resemble upscaling in the equilibrium limit, with the computational benefits that follow. In this study, the method is established as a preconditioner with good scalability properties for challenging problems regardless of dominant physics, thus laying the foundation for further studies with physical information in the preconditioner

Solution strategies for nonlinear conservation laws

Nonlinear conservation laws form the basis for models for a wide range of physical phenomena. Finding an optimal strategy for solving these problems can be challenging, and a good strategy for one problem may fail spectacularly for others. As different problems have different challenging features, exploiting knowledge about the problem structure is a key factor in achieving an efficient solution strategy. Most strategies found in literature for solving nonlinear problems involve a linearization step, usually using Newton's method, which replaces the original nonlinear problem by an iteration process consisting of a series of linear problems. A large effort is then spent on finding a good strategy for solving these linear problems. This involves choosing suitable preconditioners and linear solvers. This approach is in many cases a good choice and a multitude of different methods have been developed. However, the linearization step to some degree involves a loss of information about the original problem. This is not necessarily critical, but in many cases the structure of the nonlinear problem can be exploited to a larger extent than what is possible when working solely on the linearized problem. This may involve knowledge about dominating physical processes and specifically on whether a process is near equilibrium. By using nonlinear preconditioning techniques developed in recent years, certain attractive features such as automatic localization of computations to parts of the problem domain with the highest degree of nonlinearities arise. In the present work, these methods are further refined to obtain a framework for nonlinear preconditioning that also takes into account equilibrium information. This framework is developed mainly in the context of porous media, but in a general manner, allowing for application to a wide range of problems. A scalability study shows that the method is scalable for challenging two-phase flow problems. It is also demonstrated for nonlinear elasticity problems. Some models arising from nonlinear conservation laws are best solved using completely different strategies than the approach outlined above. One such example can be found in the field of surface gravity waves. For special types of nonlinear waves, such as solitary waves and undular bores, the well-known Korteweg-de Vries (KdV) equation has been shown to be a suitable model. This equation has many interesting properties not typical of nonlinear equations which may be exploited in the solver, and strategies usually reserved to linear problems may be applied. In this work includes a comparative study of two discretization methods with highly different properties for this equation

Design of a proteolytically stable sodium-calcium exchanger 1 activator peptide for in vivo studies

The cardiac sodium–calcium exchanger (NCX1) is important for normal Na + - and Ca 2+ -homeostasis and cardiomyocyte relaxation and contraction. It has been suggested that NCX1 activity is reduced by phosphorylated phospholemman (pSer68-PLM); however its direct interaction with PLM is debated. Disruption of the potentially inhibitory pSer68-PLM-NCX1 interaction might be a therapeutic strategy to increase NCX1 activity in cardiac disease. In the present study, we aimed to analyze the binding affinities and kinetics of the PLM-NCX1 and pSer68-PLM-NCX1 interactions by surface plasmon resonance (SPR) and to develop a proteolytically stable NCX1 activator peptide for future in vivo studies. The cytoplasmic parts of PLM (PLM cyt ) and pSer68-PLM (pSer68-PLM cyt ) were found to bind strongly to the intracellular loop of NCX1 (NCX1 cyt ) with similar K D values of 4.1 ± 1.0 nM and 4.3 ± 1.9 nM, but the PLM cyt -NCX1 cyt interaction showed higher on/off rates. To develop a proteolytically stable NCX1 activator, we took advantage of a previously designed, high-affinity PLM binding peptide (OPT) that was derived from the PLM binding region in NCX1 and that reverses the inhibitory PLM (S68D)-NCX1 interaction in HEK293. We performed N- and C-terminal truncations of OPT and identified PYKEIEQLIELANYQV as the minimum sequence required for pSer68-PLM binding. To increase peptide stability in human serum, we replaced the proline with an N-methyl-proline (NOPT) after identification of N-terminus as substitution tolerant by two-dimensional peptide array analysis. Mass spectrometry analysis revealed that the half-life of NOPT was increased 17-fold from that of OPT. NOPT pulled down endogenous PLM from rat left ventricle lysate and exhibited direct pSer68-PLM binding in an ELISA-based assay and bound to pSer68-PLM cyt with a K D of 129 nM. Excess NOPT also reduced the PLM cyt -NCX1 cyt interaction in an ELISA-based competition assay, but in line with that NCX1 and PLM form oligomers, NOPT was not able to outcompete the physical interaction between endogenous full length proteins. Importantly, cell-permeable NOPT-TAT increased NCX1 activity in cardiomyocytes isolated from both SHAM-operated and aorta banded heart failure (HF) mice, indicating that NOPT disrupted the inhibitory pSer68-PLM-NCX1 interaction. In conclusion, we have developed a proteolytically stable NCX1-derived PLM binding peptide that upregulates NCX1 activity in SHAM and HF cardiomyocytes

Design of a Proteolytically Stable Sodium-Calcium Exchanger 1 Activator Peptide for In Vivo Studies

The cardiac sodium–calcium exchanger (NCX1) is important for normal Na+- and Ca2+-homeostasis and cardiomyocyte relaxation and contraction. It has been suggested that NCX1 activity is reduced by phosphorylated phospholemman (pSer68-PLM); however its direct interaction with PLM is debated. Disruption of the potentially inhibitory pSer68-PLM-NCX1 interaction might be a therapeutic strategy to increase NCX1 activity in cardiac disease. In the present study, we aimed to analyze the binding affinities and kinetics of the PLM-NCX1 and pSer68-PLM-NCX1 interactions by surface plasmon resonance (SPR) and to develop a proteolytically stable NCX1 activator peptide for future in vivo studies. The cytoplasmic parts of PLM (PLMcyt) and pSer68-PLM (pSer68-PLMcyt) were found to bind strongly to the intracellular loop of NCX1 (NCX1cyt) with similar KD values of 4.1 ± 1.0 nM and 4.3 ± 1.9 nM, but the PLMcyt-NCX1cyt interaction showed higher on/off rates. To develop a proteolytically stable NCX1 activator, we took advantage of a previously designed, high-affinity PLM binding peptide (OPT) that was derived from the PLM binding region in NCX1 and that reverses the inhibitory PLM (S68D)-NCX1 interaction in HEK293. We performed N- and C-terminal truncations of OPT and identified PYKEIEQLIELANYQV as the minimum sequence required for pSer68-PLM binding. To increase peptide stability in human serum, we replaced the proline with an N-methyl-proline (NOPT) after identification of N-terminus as substitution tolerant by two-dimensional peptide array analysis. Mass spectrometry analysis revealed that the half-life of NOPT was increased 17-fold from that of OPT. NOPT pulled down endogenous PLM from rat left ventricle lysate and exhibited direct pSer68-PLM binding in an ELISA-based assay and bound to pSer68-PLMcyt with a KD of 129 nM. Excess NOPT also reduced the PLMcyt-NCX1cyt interaction in an ELISA-based competition assay, but in line with that NCX1 and PLM form oligomers, NOPT was not able to outcompete the physical interaction between endogenous full length proteins. Importantly, cell-permeable NOPT-TAT increased NCX1 activity in cardiomyocytes isolated from both SHAM-operated and aorta banded heart failure (HF) mice, indicating that NOPT disrupted the inhibitory pSer68-PLM-NCX1 interaction. In conclusion, we have developed a proteolytically stable NCX1-derived PLM binding peptide that upregulates NCX1 activity in SHAM and HF cardiomyocytes

Hypokalemia Promotes Arrhythmia by Distinct Mechanisms in Atrial and Ventricular Myocytes.

RationaleHypokalemia occurs in up to 20% of hospitalized patients and is associated with increased incidence of ventricular and atrial fibrillation. It is unclear whether these differing types of arrhythmia result from direct and perhaps distinct effects of hypokalemia on cardiomyocytes.ObjectiveTo investigate proarrhythmic mechanisms of hypokalemia in ventricular and atrial myocytes.Methods and resultsExperiments were performed in isolated rat myocytes exposed to simulated hypokalemia conditions (reduction of extracellular [K+] from 5.0 to 2.7 mmol/L) and supported by mathematical modeling studies. Ventricular cells subjected to hypokalemia exhibited Ca2+ overload and increased generation of both spontaneous Ca2+ waves and delayed afterdepolarizations. However, similar Ca2+-dependent spontaneous activity during hypokalemia was only observed in a minority of atrial cells that were observed to contain t-tubules. This effect was attributed to close functional pairing of the Na+-K+ ATPase and Na+-Ca2+ exchanger proteins within these structures, as reduction in Na+ pump activity locally inhibited Ca2+ extrusion. Ventricular myocytes and tubulated atrial myocytes additionally exhibited early afterdepolarizations during hypokalemia, associated with Ca2+ overload. However, early afterdepolarizations also occurred in untubulated atrial cells, despite Ca2+ quiescence. These phase-3 early afterdepolarizations were rather linked to reactivation of nonequilibrium Na+ current, as they were rapidly blocked by tetrodotoxin. Na+ current-driven early afterdepolarizations in untubulated atrial cells were enabled by membrane hyperpolarization during hypokalemia and short action potential configurations. Brief action potentials were in turn maintained by ultra-rapid K+ current (IKur); a current which was found to be absent in tubulated atrial myocytes and ventricular myocytes.ConclusionsDistinct mechanisms underlie hypokalemia-induced arrhythmia in the ventricle and atrium but also vary between atrial myocytes depending on subcellular structure and electrophysiology