Time Integration and Spatial Discretisations (Part 3)
Chair: Kees Vuik
An operator splitting approach for the time discretization of coupled systems of partial and ordinary differential equations for fluid flows
Lucia Carichino, Giovanna Guidoboni, Marcela Szopos
Abstract: Multiscale coupling of nonlinear lumped and distributes
ow models is of interest when modeling
complex hydraulic networks e.g. oil ducts, water supply, biological
uidic channels etc. In the
present work, we are motivated by the computational modeling of blood
ow through the cardiovascular
system and we focus on the multiscale coupling in terms of spatial dimensions , leading to the coupling
between partial and ordinary dierential systems.
We rst consider non-stationary Stokes systems modeling
ow of an incompressible viscous
uid in rigid
domains, coupled with nonlinear systems of ordinary dierential equations (ODEs) representing lumped
descriptions of the the
ow in dierent parts of a complex hydraulic network. Multiple connections among
Stokes domains and lumped circuits are allowed.
We propose a new splitting approach to numerically solve this multiscale problem in an ecient, accurate
and aordable manner . The main novelty of the splitting scheme is that it ensures that the energy
of the semi-discrete problem mirrors the behavior of the energy of the fully coupled problem. As a
result, unconditional stability with respect to the time step choice is obtained without the need of sub-
iterating between PDE and ODE sub-steps. We next illustrate the performances of the proposed method
by applying it to several examples in which we solve in separate blocks nonlinearities coming from
dierent sources. In particular, we derive explicit solutions for the full coupled problem in three dierent
meaningful congurations, against which our (as well as other) numerical methods can be tested. Finally
we discuss further extensions and improvements that provide increased accuracy in time or ability to
handle Navier-Stokes equations, obtained by combining the proposed scheme with other operator splitting
techniques already developed .
 Quarteroni, A., Veneziani, A., Vergara, C., Geometric multiscale modeling of the cardiovas-
cular system, between theory and practice, Computer Methods in Applied Mechanics and Engineer-
ing, 302, 193-252, 2016.
 Glowinski, R., Handbook of Numerical Analysis: Numerical methods for Fluids (Part 3), Philippe
G. Ciarlet and Jacques-Louis Lions (Eds). North-Holland, Vol. IX. (2003).
 Carichino, L., Guidoboni, G., Szopos, M., Energy-based operator splitting approach for the
time discretization of coupled systems of partial and ordinary dierential equations for
The Stokes case. Journal of Computational Physics, 364, 235-256, 2018.
Lucia CARICHINO, Worcester Polytechnic Institute, Department of Mathematical Sciences
SIGNED DISTANCE FUNCTION BASED NON-RIGID REGISTRATION OF SEQUENCES WITH VARYING IMAGE INTENSITY FUNCTION
Kateřina Solovská, Tomáš Oberhuber, Jaroslav Tintěra, Radomı́r Chabiniok
Abstract: In this talk, non-rigid registration of cardiac MR images, particularly
the Modified Look-Locker Inversion Recovery (MOLLI) sequences, will be discussed. The MOLLI sequence consists of 11 heart images acquired over 17 cardiac cycles during a single breath-hold. The images of MOLLI sequence are used for pixel-wise estimation of T1 relaxation time values. In this case the registration is necessary to correct the deformations that occur because of the patient's imperfect breath-holding during the acquisition. The main characteristic of the MOLLI sequence is the evolving intensity of the tissues and also a large variation of the image contrast. This characteristics of the sequence make the registration process challenging and make the use of intensity-based registration method impossible.
For this purpose, we propose a method based on optical flow, using information obtained by image segmentation. The first step of the registration process, is segmentation of the regions of interest, using the level set method. The segmented objects are represented by distance maps. The transformation between original images is determined by applying the optical flow method to the distance maps. The registration process is independent of the varying intensity and takes into account only the shape and position of the segmented areas, such as the myocardium or the blood pool in the ventricles.
Results of the proposed method tested on several MOLLI sequences will be presented. The results will be compared to the results of method based on maximisation of mutual information and the difference between these methods will be discussed.
Bifurcation Parameters on Cervical Cancer Mathematical Model
Tri Sri Noor Asih, Widodo Widodo, Fajar Adi Kusumo, Lina Aryati
Abstract: Abstract. The natural history of cervical cancer ilustrating the progression of cervical tissue, from normal cervic, infected by Human Papillomavirus, progress into pre-cancer and finally become cervical cancer. This proses can be representated as a dynamic system on system of ordinary differential equations. We devide the population of cervical cells into sub population of normal cells, infected cells, pre-cancer cells and cancer cells. We also include the population of Human Papillomavirus into the model. From the analysis and simulation of the system, we found some bifurcation phenomena. The existences of the equilibria and their local stability are depends on basic reproduction number and some other conditions. The basic reproduction number are depends on infection rate, the number of new virion produce by infected cells, death rate of free viruses, growth rate of infected cells, and progression rate of pre-cancer cells become cancer cells. So we suspect that some of those parameters will become a bifurcation parameters. We do some simulation by Auto to detect the bifurcation, and connecting with the analysis result before. By continuation of infection rate we got some fold bifurcation. And while we made continuation on infection rate together with invasion rate we found cusp bifurcation.
Well-Balanced and Asymptotic Preserving IMEX-Peer Methods
Moritz Schneider, Jens Lang
Abstract: Peer methods are a comprehensive class of time integrators offering numerous degrees of freedom in their coefficient matrices that can be used to ensure advantageous properties, e.g. A-stability or super-convergence. In this paper, we show that implicit-explicit (IMEX) Peer methods are well-balanced and asymptotic preserving by construction without additional constraints on the coefficients. These properties are relevant when solving (the space discretisation of) hyperbolic systems of balance laws, for example. Numerical examples confirm the theoretical results and illustrate the potential of IMEX-Peer methods.