There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

https://depts.washington.edu/clawpack/book2/sample.pdf

Finite Volume Methods for Hyperbolic Problems

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

高等学校計算数学学報 = Numerical mathematics

The broad research thematic of flows on networks was addressed in recent years by many researchers, in the area of applied mathematics, with new models based on partial differential equations. The latter brought a significant innovation in a field previously dominated by more classical techniques from discrete mathematics or methods based on ordinary differential equations. In particular, a number of results, mainly dealing with vehicular traffic, supply chains and data networks, were collected in two monographs: Traffic flow on networks, AIMSciences, Springfield, 2006, and Modeling, simulation, and optimization of supply chains, SIAM, Philadelphia, 2010. The field continues to flourish and a considerable number of papers devoted to the subject is published every year, also because of the wide and increasing range of applications: from blood flow to air traffic management. The aim of the present survey paper is to provide a view on a large number of themes, results and applications related to this broad research direction. The authors cover different expertise (modeling, analysis, numeric, optimization and other) so to provide an overview as extensive as possible. The focus is mainly on developments which appeared subsequently to the publication of the aforementioned books.

https://www.ems-ph.org/journals/show_pdf.php?issn=2308-2151&vol=1&iss=1&rank=2

Flows on networks: recent results and perspectives

https://www.math.uh.edu/~canic/hemopapers/FINAL_JCP_PAPER.pdf

Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow

In this paper, we present a mathematical analysis of the quasilinear effects arising in a hyperbolic system of partial differential equations modelling blood flow through large compliant vessels. The equations are derived using asymptotic reduction of the incompressible Navier–Stokes equations in narrow, long channels.



To guarantee strict hyperbolicity we first derive the estimates on the initial and boundary data which imply strict hyperbolicity in the region of smooth flow. We then prove a general theorem which provides conditions under which an initial–boundary value problem for a quasilinear hyperbolic system admits a smooth solution. Using this result we show that pulsatile flow boundary data always give rise to shock formation (high gradients in the velocity and inner vessel radius). We estimate the time and the location of the first shock formation and show that in a healthy individual, shocks form well outside the physiologically interesting region (2.8m downstream from the inlet boundary). In the end we present a study of the influence of vessel tapering on shock formation. We obtain a surprising result: vessel tapering postpones shock formation. We provide an explanation for why this is the case. Copyright © 2003 John Wiley & Sons, Ltd.

Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels

We study a nonlinear, unsteady, moving boundary, fluid–structure interaction (FSI) problem arising in modeling blood flow through elastic and viscoelastic arteries. The fluid flow, which is driven by the time-dependent pressure data, is governed by two-dimensional incompressible Navier–Stokes equations, while the elastodynamics of the cylindrical wall is modeled by the one-dimensional cylindrical Koiter shell model. Two cases are considered: the linearly viscoelastic and the linearly elastic Koiter shell. The fluid and structure are fully coupled (two-way coupling) via the kinematic and dynamic lateral boundary conditions describing continuity of velocity (the no-slip condition), and the balance of contact forces at the fluid–structure interface. We prove the existence of weak solutions to the two FSI problems (the viscoelastic and the elastic case) as long as the cylinder radius is greater than zero. The proof is based on a novel semi-discrete, operator splitting numerical scheme, known as the kinematically coupled scheme, introduced in Guidoboni et al. (J Comput Phys 228(18):6916–6937, 2009) to numerically solve the underlying FSI problems. The backbone of the kinematically coupled scheme is the well-known Marchuk–Yanenko scheme, also known as the Lie splitting scheme. We effectively prove convergence of that numerical scheme to a solution of the corresponding FSI problem.

Existence of a Weak Solution to a Nonlinear Fluid–Structure Interaction Problem Modeling the Flow of an Incompressible, Viscous Fluid in a Cylinder with Deformable Walls

We prove the existence of a solution of a free boundary problem for the transonic small-disturbance equation The free boundary is the position of a transonic shock dividing two regions of smooth flow Assuming inviscid, irrotational flow, as modeled by the transonic small-disturbance equation, the equation is hyperbolic upstream where the flow is supersonic, and elliptic in the downstream subsonic region To study the stability of a uniform planar transonic shock, we consider perturbation by a steady C1+ϵ upstream disturbance If the upstream disturbance is small in a C1 sense, then there is a steady solution in which the downstream flow and the transonic shock are Holder-continuous perturbations of the uniform configuration This result provides a new use of inviscid perturbation techniques to demonstrate, in two dimensions, the stability of transonic shock waves of the type that appear, for example, over the wing of an airplane, along an airfoil, or as bow shocks in a flow with a supersonic free-stream velocity © 2000 John Wiley & Sons, Inc

Sunčica Čanić

Papers

Flows on networks: recent results and perspectives

Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow

Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels

Existence of a Weak Solution to a Nonlinear Fluid–Structure Interaction Problem Modeling the Flow of an Incompressible, Viscous Fluid in a Cylinder with Deformable Walls

A proof of existence of perturbed steady transonic shocks via a free boundary problem