No Arabic abstract
The present work was inspired by the recent developments in laboratory experiments made on chip, where culturing of multiple cell species was possible. The model is based on coupled reaction-diffusion-transport equations with chemotaxis, and takes into account the interactions among cell populations and the possibility of drug administration for drug testing effects. Our effort was devoted to the development of a simulation tool that is able to reproduce the chemotactic movement and the interactions between different cell species (immune and cancer cells) living in microfluidic chip environment. The main issues faced in this work are the introduction of mass-preserving and positivity-preserving conditions involving the balancing of incoming and outgoing fluxes passing through interfaces between 2D and 1D domains of the chip and the development of mass-preserving and positivity preserving numerical conditions at the external boundaries and at the interfaces between 2D and 1D domains.
In this paper, we present an interpolation framework for structure-preserving model order reduction of parametric bilinear dynamical systems. We introduce a general setting, covering a broad variety of different structures for parametric bilinear systems, and then provide conditions on projection spaces for the interpolation of structured subsystem transfer functions such that the system structure and parameter dependencies are preserved in the reduced-order model. Two benchmark examples with different parameter dependencies are used to demonstrate the theoretical analysis.
It is well-known that a numerical method which is at the same time geometric structure-preserving and physical property-preserving cannot exist in general for Hamiltonian partial differential equations. In this paper, we present a novel class of parametric multi-symplectic Runge-Kutta methods for Hamiltonian wave equations, which can also conserve energy simultaneously in a weaker sense with a suitable parameter. The existence of such a parameter, which enforces the energy-preserving property, is proved under certain assumptions on the fixed step sizes and the fixed initial condition. We compare the proposed method with the classical multi-symplectic Runge-Kutta method in numerical experiments, which shows the remarkable energy-preserving property of the proposed method and illustrate the validity of theoretical results.
We propose a new Lagrange multiplier approach to construct positivity preserving schemes for parabolic type equations. The new approach introduces a space-time Lagrange multiplier to enforce the positivity with the Karush-Kuhn-Tucker (KKT) conditions. We then use a predictor-corrector approach to construct a class of positivity schemes: with a generic semi-implicit or implicit scheme as the prediction step, and the correction step, which enforces the positivity, can be implemented with negligible cost. We also present a modification which allows us to construct schemes which, in addition to positivity preserving, is also mass conserving. This new approach is not restricted to any particular spatial discretization and can be combined with various time discretization schemes. We establish stability results for our first- and second-order schemes under a general setting, and present ample numerical results to validate the new approach.
In this paper, a family of arbitrarily high-order structure-preserving exponential Runge-Kutta methods are developed for the nonlinear Schrodinger equation by combining the scalar auxiliary variable approach with the exponential Runge-Kutta method. By introducing an auxiliary variable, we first transform the original model into an equivalent system which admits both mass and modified energy conservation laws. Then applying the Lawson method and the symplectic Runge-Kutta method in time, we derive a class of mass- and energy-preserving time-discrete schemes which are arbitrarily high-order in time. Numerical experiments are addressed to demonstrate the accuracy and effectiveness of the newly proposed schemes.
In the framework of accurate and efficient segregated schemes for 3D cardiac electromechanics and 0D cardiovascular models, we propose here a novel numerical approach to address the coupled 3D-0D problem introduced in Part I of this two-part series of papers. We combine implicit-explicit schemes to solve the different cardiac models in a multiphysics setting. We properly separate and manage the different time and space scales related to cardiac electromechanics and blood circulation. We employ a flexible and scalable intergrid transfer operator that enables to interpolate Finite Element functions among different meshes and, possibly, among different Finite Element spaces. We propose a numerical method to couple the 3D electromechanical model and the 0D circulation model in a numerically stable manner within a fully segregated fashion. No adaptations are required through the different phases of the heartbeat. We also propose a robust algorithm to reconstruct the stress-free reference configuration. Due to the computational cost associated with the numerical solution of this inverse problem, the reference configuration recovery algorithm comes along with a novel projection technique to precisely recover the unloaded geometry from a coarser representation of the computational domain. We show the convergence property of our numerical schemes by performing an accuracy study through grid refinement. To prove the biophysical accuracy of our computational model, we also address different scenarios of clinical interest in our numerical simulations by varying preload, afterload and contractility. Indeed, we simulate physiologically relevant behaviors and we reproduce meaningful results in the context of cardiac function.