Susana Serna

A characteristic-based nonconvex entropy-fix upwind scheme for the ideal magnetohydrodynamic equations

In this paper we perform an analysis of the wave structure of the ideal magnetohydrodynamic (MHD) equations. We present an analytical expression of the nonlinearity term associated to each characteristic field derived from a scaled version of the complete system of eigenvectors proposed by Brio and Wu [M. Brio, C.C. Wu, An upwind differencing scheme for the equations of ideal magnetohydrodynamics, J. Comput. Phys. 75 (2) (1988) 400-422] and adopting the eight wave approach by Powell et al. [K.G. Powell, P.L. Roe, R.S. Myong, T. Gombosi, D. deZeeuw, An upwind scheme for magnetohydrodynamics, AIAA 12th Computational Fluid Dynamics Conference, San Diego, CA, 1995, pp. 661-674]. A criterion for the detection of local regions containing points for which a nonlinear characteristic field becomes nonconvex is formulated for the two-dimensional case. We then design a characteristic-based upwind scheme for the ideal MHD equations that resolves the wave dynamics by local characteristic wavefields. The new scheme is able to detect local regions containing nonconvex singularities and to handle an entropy correction through a prescription of a local viscosity ensuring convergence to the entropy solution. A third order accurate version of the scheme performs satisfactorily in resolving one and two-dimensional MHD problems. Numerical results indicate that the proposed scheme behaves low dissipative, stable and accurate under high CFL numbers.

A local spectral inversion of a linearized TV model for denoising and deblurring

In this paper, we propose a model for denoising and deblurring consisting of a system of linear partial differential equations with locally constant coefficients, obtained as a local linearization of the total variation models. The keypoint of our model is to get the local inversion of the Laplacian operator, which will be done via the Fast Fourier Transform (FFT). Two local schemes will be developed: a pointwise and a piecewise one. We will analyze both, their advantages and their limitations.

Diffusion and reaction in the cell glycocalyx and the extracellular matrix

Many biologically important macromolecular reactions are assembled and catalyzed at the cell lipid-surface and thus, the extracellular matrix and the glycocalyx layer mediate transfer and exchange of reactants and products between the flowing blood and the catalytic lipid-surface. This paper presents a mathematical model of reaction–diffusion equations that simply describes the transfer process and explores its influence on surface reactivity for a prototypical pathway, the tissue factor (Tf) pathway of blood coagulation. The progressively increasing friction offered by the matrix and glycocalyx to reactants and to the product (coagulation factors X, VIIa and Xa) approaching the reactive surface is simulated and tested by solving the equations numerically with both, monotonically decreasing and constant diffusion profiles. Numerical results show that compared to isotropic transfer media, the anisotropic structure of the matrix and glycocalyx sharply decreases overall reaction rates and significantly increases the mean transit time of reactants; this implies that the anisotropy modifies the distribution of reactants. Results also show that the diffusional transfer, whether isotropic or anisotropic, influences reaction rates according to the order at which the reactants arrive at the boundary. Faster rates are observed when at least one of the reactants is homogeneously distributed before the other arrives at the boundary than when both reactants transfer simultaneously from the boundary.

Anomalous wave structure in magnetized materials described by non-convex equations of state

We analyze the anomalous wave structure appearing in flow dynamics under the influence of magnetic field in materials described by non-ideal equations of state. We consider the system of magnetohydrodynamics equations closed by a general equation of state (EOS) and propose a complete spectral decomposition of the fluxes that allows us to derive an expression of the nonlinearity factor as the mathematical tool to determine the nature of the wave phenomena. We prove that the possible formation of non-classical wave structure is determined by both the thermodynamic properties of the material and the magnetic field as well as its possible rotation. We demonstrate that phase transitions induced by material properties do not necessarily imply the loss of genuine nonlinearity of the wavefields as is the case in classical hydrodynamics. The analytical expression of the nonlinearity factor allows us to determine the specific amount of magnetic field necessary to prevent formation of complex structure induced by phase transition in the material. We illustrate our analytical approach by considering two non-convex EOS that exhibit phase transitions and anomalous behavior in the evolution. We present numerical experiments validating the analysis performed through a set of one-dimensional Riemann problems. In the examples we show how to determine the appropriate amount of magnetic field in the initial conditions of the Riemann problem to transform a thermodynamic composite wave into a simple nonlinear wave.

Numerical solution of the quantum Lenard-Balescu equation for a non-degenerate one-component plasma

Summary form only given. The quantum Lenard-Balescu (QLB) equation provides an accurate description of weakly-couple plasmas. Unlike Landau/Fokker-Planck, the QLB equation is fully convergent and thus requires no input Coulomb logarithm. However, it contains a dielectric function that depends on the distribution, leading to a very complicated integro-differential equation. We present what we believe is the first numerical solution of the quantum Lenard-Balescu equation. We use a spectral expansion in Laguerre polynomials, which gives automatic conservation of particles and energy and enables an accurate integration over the dielectric function. Our method can also be used to solve the Landau/Fokker-Planck equation and we present comparisons with the QLB system for various equilibration problems.

Analysis of unstable behavior in a mathematical model for erythropoiesis

We consider an age-structured model that describes the regulation of erythropoiesis through the negative feedback loop between erythropoietin and hemoglobin. This model is reduced to a system of two ordinary differential equations with two constant delays for which we show existence of a unique steady state. We determine all instances at which this steady state loses stability via a Hopf bifurcation through a theoretical bifurcation analysis establishing analytical expressions for the scenarios in which they arise. We show examples of supercritical Hopf bifurcations for parameter values estimated according to physiological values for humans found in the literature and present numerical simulations in agreement with the theoretical analysis. We provide a strategy for parameter estimation to match empirical measurements and predict dynamics in experimental settings, and compare existing data on hemoglobin oscillation in rabbits with predictions of our model.

Afternotes on PHM: Harmonic ENO Methods

PHM methods have been used successfully as reconstruction procedures to design high-order Riemann solvers for nonlinear scalar and systems of conservation laws, (see [8], [1], [4]). We introduce a new class of polynomial reconstruction procedures based on the harmonic mean of the absolute values of finite diferences used as difference-limiter, following the original idea used before to design the piecewise hyperbolic method, introduced in [8]. We call those methods ’harmonic ENO methods’, (HENO). Furthermore, we give analytical and numerical evidence of the good behavior of these methods used as reconstruction procedures for the numerical approximation by means of shock-capturing methods for scalar and systems of conservation laws in ID. We discuss, in particular, the behavior of a fourth order harmonic ENO method,(HEN04 in short), compared with PHM, EN03 and WEN05 methods, (see [2], [10], [3]).

Numerical solution of the quantum Lenard-Balescu equation

The quantum Lenard-Balescu (QLB) equation provides an accurate description of weakly-couple plasmas. Unlike Landau/Fokker-Planck, the QLB equation is fully convergent and thus requires no input Coulomb logarithm. However, it contains a dielectric function that depends on the distribution, leading to a very complicated integro-differential equation. We present what we believe is the first numerical solution of the quantum Lenard-Balescu equation. We use a spectral expansion in Laguerre polynomials, which gives automatic conservation of particles and energy and enables an accurate integration over the dielectric function. Our method can also be used to solve the Landau/Fokker-Planck equation and we present comparisons with the QLB system for various equilibration problems.

Analysis and numerical approximation of viscosity solutions with shocks: application to the plasma equation

We consider a new class of Hamilton‐Jacobi equations arising from the convective part of general Fokker‐Planck equations ruled by a non‐negative diffusion function that depends on the unknown and on the gradient of the unknown. The new class of Hamilton‐Jacobi equations represents the propagation of fronts with speed that is a nonlinear function of the signal. The equations contain a nonstandard Hamiltonian that allows the presence of shocks in the solution and these shocks propagate with nonlinear velocity. We focus on the one‐dimensional plasma equation as an example of the general Fokker‐Planck equations having the features we are interested in analyzing. We explore features of the solution of the corresponding Hamilton‐Jacobi plasma equation and propose a suitable fifth order finite difference numerical scheme that approximates the solution in a consistent way with respect to the solution of the associated Fokker‐Planck equation. We present numerical results performed under different initial data with...

High order accurate shock capturing schemes for two-component Richtmyer-Meshkov instabilities in compressible magnetohydrodynamics

We design a conservative and entropy satisfying numerical scheme to perform numerical simulations of two component Richtmyer-Meshkov (RM) instabilities in compressible magnetohydrodynamics (MHD). We first formulate a conservative model of a two-component compressible MHD fluid ruled under two ideal gases with different adiabatic exponents. The formulation includes a level set function that allows to evolve the two components of the plasma in a conservative and consistent way. We present a set of examples including two-component Riemann problems and high Mach shock wave interactions with entropy contact waves that validate the high order accurate numerical scheme. We observe that turbulent regimes are completely developed in different examples where shocks, contacts and rarefactions waves propagate with correct speed.