Well-balanced high-order numerical schemes for one-dimensional blood flow in vessels with varying mechanical properties

We construct well-balanced, high-order numerical schemes for one-dimensional blood flow in elastic vessels with varying mechanical properties. We adopt the ADER (Arbitrary high-order DERivatives) finite volume framework, which is based on three building blocks: a first-order monotone numerical flux, a non-linear spatial reconstruction operator and the solution of the Generalised (or high-order) Riemann Problem. Here, we first construct a well-balanced first-order numerical flux following the Generalised Hydrostatic Reconstruction technique. Then, a conventional non-linear spatial reconstruction operator and the local solver for the Generalised Riemann Problem are modified in order to preserve well-balanced properties. A carefully chosen suit of test problems is used to systematically assess the proposed schemes and to demonstrate that well-balanced properties are mandatory for obtaining correct numerical solutions for both steady and time-dependent problems.

[1]  Manuel Jesús Castro Díaz,et al.  Well-Balanced High Order Extensions of Godunov's Method for Semilinear Balance Laws , 2008, SIAM J. Numer. Anal..

[2]  Michael Dumbser,et al.  Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws , 2008, J. Comput. Phys..

[3]  Pablo J. Blanco,et al.  Assessing the influence of heart rate in local hemodynamics through coupled 3D‐1D‐0D models , 2010 .

[4]  Carlos Parés Madroñal,et al.  On the Convergence and Well-Balanced Property of Path-Conservative Numerical Schemes for Systems of Balance Laws , 2011, J. Sci. Comput..

[5]  Randall J. LeVeque,et al.  Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods , 1998 .

[6]  Manuel J. Castro,et al.  WELL-BALANCED NUMERICAL SCHEMES BASED ON A GENERALIZED HYDROSTATIC RECONSTRUCTION TECHNIQUE , 2007 .

[7]  M. J. Castro,et al.  FORCE schemes on unstructured meshes II: Non-conservative hyperbolic systems , 2010 .

[8]  Michael Dumbser,et al.  Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms , 2012, J. Comput. Phys..

[9]  Timothy J. Pedley,et al.  Numerical solutions for unsteady gravity-driven flows in collapsible tubes: evolution and roll-wave instability of a steady state , 1999, Journal of Fluid Mechanics.

[10]  M. J. Castro,et al.  ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows , 2009 .

[11]  Alfredo Bermúdez,et al.  Upwind methods for hyperbolic conservation laws with source terms , 1994 .

[12]  Emmanuel Audusse,et al.  A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows , 2004, SIAM J. Sci. Comput..

[13]  Michael Dumbser,et al.  ADER Schemes for Nonlinear Systems of Stiff Advection–Diffusion–Reaction Equations , 2011, J. Sci. Comput..

[14]  Alfio Quarteroni,et al.  Cardiovascular mathematics : modeling and simulation of the circulatory system , 2009 .

[15]  C. Parés Numerical methods for nonconservative hyperbolic systems: a theoretical framework. , 2006 .

[16]  Michael Dumbser,et al.  A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes , 2008, J. Comput. Phys..

[17]  Rupert Klein,et al.  Well balanced finite volume methods for nearly hydrostatic flows , 2004 .

[18]  Eleuterio F. Toro,et al.  Centred TVD schemes for hyperbolic conservation laws , 2000 .

[19]  Eleuterio F. Toro,et al.  Simplified blood flow model with discontinuous vessel properties: Analysis and exact solutions , 2012 .

[20]  George Em Karniadakis,et al.  Simulation of the human intracranial arterial tree. , 2009, Philosophical transactions. Series A, Mathematical, physical, and engineering sciences.

[21]  Eleuterio F. Toro,et al.  Flow in Collapsible Tubes with Discontinuous Mechanical Properties: Mathematical Model and Exact Solutions , 2013 .

[22]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[23]  Alfio Quarteroni,et al.  A One Dimensional Model for Blood Flow: Application to Vascular Prosthesis , 2002 .

[24]  A. Quarteroni,et al.  One-dimensional models for blood flow in arteries , 2003 .

[25]  Eleuterio F. Toro,et al.  Towards Very High Order Godunov Schemes , 2001 .

[26]  S. Sherwin,et al.  Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vitro measurements , 2011, Journal of biomechanics.

[27]  Manuel Jesús Castro Díaz,et al.  Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes , 2008, J. Comput. Phys..

[28]  E. Toro,et al.  Solution of the generalized Riemann problem for advection–reaction equations , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[29]  G. D. Maso,et al.  Definition and weak stability of nonconservative products , 1995 .