Gino I. Montecinos

Hyperbolic reformulation of a 1D viscoelastic blood flow model and ADER finite volume schemes

The applicability of ADER finite volume methods to solve hyperbolic balance laws with stiff source terms in the context of well-balanced and non-conservative schemes is extended to solve a one-dimensional blood flow model for viscoelastic vessels, reformulated as a hyperbolic system, via a relaxation time. A criterion for selecting relaxation times is found and an empirical convergence rate assessment is carried out to support this result. The proposed methodology is validated by applying it to a network of viscoelastic vessels for which experimental and numerical results are available. The agreement between the results obtained in the present paper and those available in the literature is satisfactory. Key features of the present formulation and numerical methodologies, such as accuracy, efficiency and robustness, are fully discussed in the paper.

Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms

We compare four different approximate solvers for the generalized Riemann problem (GRP) for non-linear systems of hyperbolic equations with source terms. The GRP is a special Cauchy problem for a hyperbolic system with source terms whose initial condition is piecewise smooth. We briefly review the four solvers currently available and carry out a systematic assessment of these in terms of accuracy and computational efficiency. These solvers are the building block for constructing high-order numerical schemes of the ADER type for solving the general initial-boundary value problem for inhomogeneous systems in multiple space dimensions, in the frameworks of finite volume and discontinuous Galerkin finite element methods.

Reformulations for general advection-diffusion-reaction equations and locally implicit ADER schemes

Following Cattaneos original idea, in this article we first present two relaxation formulations for time-dependent, non-linear systems of advection-diffusion-reaction equations. Such formulations yield time-dependent non-linear hyperbolic balance laws with stiff source terms. Then we present a locally implicit version of the ADER method to solve these stiff systems to high accuracy. The new ingredient of the numerical methodology is a locally implicit solution of the generalised Riemann problem. We illustrate the formulations and the resulting numerical approach by solving the compressible Navier-Stokes equations.

Advection-Diffusion-Reaction Equations: Hyperbolization and High-Order ADER Discretizations

The purpose of this paper is twofold. First, we extend the applicability of Cattaneos relaxation approach, one of the currently known relaxation approaches, to reformulate time-dependent advection...

Exploring various flux vector splittings for the magnetohydrodynamic system

In this paper we explore flux vector splittings for the MHD system of equations. Our approach follows the strategy that was initially put forward in Toro and Vazquez-Cendon (2012) 55. We split the flux vector into an advected sub-system and a pressure sub-system. The eigenvalues and eigenvectors of the split sub-systems are then studied for physical suitability. Not all flux vector splittings for MHD yield physically meaningful results. We find one that is completely useless, another that is only marginally useful and one that should work well in all regimes where the MHD equations are used. Unfortunately, this successful flux vector splitting turns out to be different from the Zha-Bilgen flux vector splitting. The eigenvalues and eigenvectors of this favorable FVS are explored in great detail in this paper.The pressure sub-system holds the key to finding a successful flux vector splitting. The eigenstructure of the successful flux vector splitting for MHD is thoroughly explored and orthonormalized left and right eigenvectors are explicitly catalogued. We present a novel approach to the solution of the Riemann problem formed by the pressure sub-system for the MHD equations. Once the pressure sub-system is solved, the advection sub-system follows naturally. Our method also works very well for the Euler system.Our FVS successfully captures isolated, stationary contact discontinuities in MHD. However, we explain why any FVS for MHD is not adept at capturing isolated, stationary Alfvenic discontinuities. Several stringent one-dimensional Riemann problems are presented to show that the method works successfully and can effectively capture the full panoply of wave structures that arise in MHD. This includes compound waves and switch-on and switch-off shocks that arise because of the non-convex nature of the MHD system.

Computational haemodynamics in stenotic internal jugular veins

An association of stenotic internal jugular veins (IJVs) to anomalous cerebral venous hemodynamics and Multiple Sclerosis has been recently hypothesized. In this work, we set up a computational framework to assess the relevance of IJV stenoses through numerical simulation, combining medical imaging, patient-specific data and a mathematical model for venous occlusions. Coupling a three-dimensional description of blood flow in IJVs with a reduced one-dimensional model for major intracranial veins, we are able to model different anatomical configurations, an aspect of importance to understand the impact of IJV stenosis in intracranial venous haemodynamics. We investigate several stenotic configurations in a physiologic patient-specific regime, quantifying the effect of the stenosis in terms of venous pressure increase and wall shear stress patterns. Simulation results are in qualitative agreement with reported pressure anomalies in pathological cases. Moreover, they demonstrate the potential of the proposed multiscale framework for individual-based studies and computer-aided diagnosis.

Implicit, semi-analytical solution of the generalized Riemann problem for stiff hyperbolic balance laws

We present a semi-analytical, implicit solution to the generalized Riemann problem (GRP) for non-linear systems of hyperbolic balance laws with stiff source terms. The solution method is based on an implicit, time Taylor series expansion and the Cauchy-Kowalewskaya procedure, along with the solution of a sequence of classical Riemann problems. Our new GRP solver is then used to construct locally implicit ADER methods of arbitrary accuracy in space and time for solving the general initial-boundary value problem for non-linear systems of hyperbolic balance laws with stiff source terms. Analysis of the method for model problems is carried out and empirical convergence rate studies for suitable tests problems are performed, confirming the theoretically expected high order of accuracy.

Junction-Generalized Riemann Problem for stiff hyperbolic balance laws in networks

In this paper we design a new implicit solver for the Junction-Generalized Riemann Problem (J-GRP), which is based on a recently proposed implicit method for solving the Generalized Riemann Problem (GRP) for systems of hyperbolic balance laws. We use the new J-GRP solver to construct an ADER scheme that is globally explicit, locally implicit and with no theoretical accuracy barrier, in both space and time. The resulting ADER scheme is able to deal with stiff source terms and can be applied to non-linear systems of hyperbolic balance laws in domains consisting on networks of one-dimensional sub-domains. In this paper we specifically apply the numerical techniques to networks of blood vessels. We report on a test problem with exact solution for a simplified network of three vessels meeting at a single junction, which is then used to carry out a systematic convergence rate study of the proposed high-order numerical methods. Schemes up to fifth order of accuracy in space and time are implemented and tested. We then show the ability of the ADER scheme to deal with stiff sources through a numerical simulation in a network of vessels. An application to a physical test problem consisting of a network of 37 compliant silicon tubes (arteries) and 21 junctions, reveals that it is imperative to use high-order methods at junctions, in order to preserve the desired high order of accuracy in the full computational domain. For example, it is demonstrated that a second-order method throughout, gives comparable results to a method that is fourth order in the interior of the domain and first order at junctions. We are concerned with stiff hyperbolic balance laws in networks.An implicit solver for the junction-generalized Riemann problem is proposed.ADER schemes of arbitrary accuracy in space and time for networks are constructed.Convergence rates studies of schemes up to fifth order are carried out.It is necessary to match the accuracy at junctions to that of the rest of the domain.

Computational electrodynamics in material media with constraint-preservation, multidimensional Riemann solvers and sub-cell resolution – Part I, second-order FVTD schemes

Abstract While classic finite-difference time-domain (FDTD) solutions of Maxwells equations have served the computational electrodynamics (CED) community very well, formulations based on Godunov methodology have begun to show advantages. We argue that the formulations presented so far are such that FDTD schemes and Godunov-based schemes each have their own unique advantages. However, there is currently not a single formulation that systematically integrates the strengths of both these major strains of development. While an early glimpse of such a formulation was offered in Balsara et al. [16] , that paper focused on electrodynamics in plasma. Here, we present a synthesis that integrates the strengths of both FDTD and Godunov-based schemes into a robust single formulation for CED in material media. Three advances make this synthesis possible. First, from the FDTD method, we retain (but somewhat modify) a spatial staggering strategy for the primal variables. This provides a beneficial constraint preservation for the electric displacement and magnetic induction vector fields via reconstruction methods that were initially developed in some of the first authors papers for numerical magnetohydrodynamics (MHD). Second, from the Godunov method, we retain the idea of upwinding, except that this idea, too, has to be significantly modified to use the multi-dimensionally upwinded Riemann solvers developed by the first author. Third, we draw upon recent advances in arbitrary derivatives in space and time (ADER) time-stepping by the first author and his colleagues. We use the ADER predictor step to endow our method with sub-cell resolving capabilities so that the method can be stiffly stable and resolve significant sub-cell variation in the material properties within a zone. Overall, in this paper, we report a new scheme for numerically solving Maxwells equations in material media, with special attention paid to a second-order-accurate formulation. Several numerical examples are presented to show that the proposed technique works. Because of its sub-cell resolving ability, the new method retains second-order accuracy even when material permeability and permittivity vary by an order-of-magnitude over just one or two zones. Furthermore, because the new method is also unconditionally stable in the presence of stiff source terms (i.e., in problems involving giant conductivity variations), it can handle several orders-of-magnitude variation in material conductivity over just one or two zones without any reduction of the time–step. Consequently, the CFL depends only on the propagation speed of light in the medium being studied.

Assessment of reduced‐order unscented Kalman filter for parameter identification in one‐dimensional blood flow models using experimental data

This work presents a detailed investigation of a parameter estimation approach on the basis of the reduced-order unscented Kalman filter (ROUKF) in the context of 1-dimensional blood flow models. In particular, the main aims of this study are (1) to investigate the effects of using real measurements versus synthetic data for the estimation procedure (i.e., numerical results of the same in silico model, perturbed with noise) and (2) to identify potential difficulties and limitations of the approach in clinically realistic applications to assess the applicability of the filter to such setups. For these purposes, the present numerical study is based on a recently published in vitro model of the arterial network, for which experimental flow and pressure measurements are available at few selected locations. To mimic clinically relevant situations, we focus on the estimation of terminal resistances and arterial wall parameters related to vessel mechanics (Youngs modulus and wall thickness) using few experimental observations (at most a single pressure or flow measurement per vessel). In all cases, we first perform a theoretical identifiability analysis on the basis of the generalized sensitivity function, comparing then the results owith the ROUKF, using either synthetic or experimental data, to results obtained using reference parameters and to available measurements.