Eleuterio F. Toro

Christian Contarino, Eleuterio F. Toro

We propose a one-dimensional model for collecting lymphatics coupled with a novel Electro-Fluid-Mechanical Contraction (EFMC) model for dynamical contractions, based on a modified FitzHugh-Nagumo model for action potentials. The one-dimensional model for a compliant lymphatic vessel is a set of hyperbolic Partial Differential Equations (PDEs). The EFMC model combines the electrical activity of lymphangions (action potentials) with fluid-mechanical feedback (stretch of the lymphatic wall and wall shear stress) and the mechanical variation of the lymphatic wall properties (contractions). The EFMC model is governed by four Ordinary Differential Equations (ODEs) and phenomenologically relies on: (1) environmental calcium influx, (2) stretch-activated calcium influx, and (3) contraction inhibitions induced by wall shear stresses. We carried out a complete mathematical analysis of the stability of the stationary state of the EFMC model. Overall, the EFMC model allows imitating the influence of pressure and wall shear stress on the frequency of contractions observed experimentally. Lymphatic valves are modelled using a well-established lumped-parameter model which allows simulating stenotic and regurgitant valves. We analysed several lymphodynamical indexes of a single lymphangion for a wide range of upstream and downstream pressure combinations. Stenotic and regurgitant valves were modelled, and their effects are here quantified. Results for stenotic valves showed in the downstream lymphangion that for low frequencies of contractions the Calculated Pump Flow (CPF) index remained almost unaltered, while for high frequencies the CPF dramatically decreased depending on the severity of the stenosis (up to 93% for a severe stenosis). Results for incompetent valves showed that the net flow during a lymphatic cycle tends to zero as the degree of incompetence increases.

Dante Kalise, Ivar Lie, Eleuterio F. Toro

We present a numerical scheme for the solution of a class of atmospheric models where high horizontal resolution is required while a coarser vertical structure is allowed. The proposed scheme considers a layering procedure for the original set of equations, and the use of high-order ADER finite volume schemes for the solution of the system of balance laws arising from the dimensional reduction procedure. We present several types of layering based upon Galerkin discretizations of the vertical structure, and we study the effect of incrementing the order of horizontal approximation. Numerical experiments for the computational validation of the convergence of the scheme together with the study of physical phenomena are performed over 2D linear advective models, including a set of equations for an isothermal atmosphere.

Eleuterio F. Toro

These personal reminiscences of the great Russian mathematician Sergey K. Godunov (1929-2023) arose from a request by his daughter, Ekaterina, to contribute a piece to a book she is writing about her father's life. I was honoured to accept this invitation and to give written form to the rewarding experience of conducting research on themes pioneered by Professor Godunov, interacting with him personally on several memorable occasions, and helping to establish research collaboration with his Novosibirsk group. Our association began at a conference in Lake Tahoe (USA) in 1995 and was followed by a number of subsequent meetings, notably in Novosibirsk, Manchester, Oxford, and Cambridge. Briefer encounters also took place in the Porquerolles Island (France), in Lyon (France), and in St. Petersburg (Russia). These notes bear witness to the global impact of Godunov's mathematical creativity across multiple branches of science, as well as to its lasting influence on the careers of generations of mathematicians in both academia and industry.

Francesca Bellamoli, Lucas Omar Müller, Eleuterio Francisco Toro

There is growing interest in developing mathematical models and appropriate numerical methods for problems involving networks formed by, essentially, one-dimensional (1D) domains joined by junctions. Examples include hyperbolic equations in networks of gas tubes, water channels and vessel networks for blood and lymph in the human circulatory system. A key point in designing numerical methods for such applications is the treatment of junctions, i.e. points at which two or more 1D domains converge and where the flow exhibits multidimensional behaviour. This paper focuses on the design of methods for networks of water channels. Our methods adopt the finite volume approach to make full use of the two-dimensional shallow water equations on the true physical domain, locally at junctions, while solving the usual one-dimensional shallow water equations away from the junctions. In addition to mass conservation, our methods enforce conservation of momentum at junctions; the latter seems to be the missing element in methods currently available. Apart from simplicity and robustness, the salient feature of the proposed methods is their ability to successfully deal with transcritical and supercritical flows at junctions, a property not enjoyed by existing published methodologies. Systematic assessment of the proposed methods for a variety of flow configurations is carried out. The methods are directly applicable to other systems, provided the multidimensional versions of the 1D equations are available.