arXiv subjects

Lucas O. Müller

Publications and source records attributed to Lucas O. Müller.

4 recordsLinked to original sources

arXiv2026

A generalized Riemann problem-based compact reconstruction method for finite volume schemes

We present a Generalized Riemann Problem-based reconstruction method (GRPrec) for high-order finite volume schemes applied to hyperbolic partial differential equations. The method constructs spatial polynomials using cell averages at the current time level and GRP solution data from the previous time level. The resulting GRPrec stencil is as compact as that of discontinuous Galerkin (DG) schemes but unlike DG, our finite volume schemes obey a generous CFL stability condition that is independent of the order of accuracy. We assess the method's performance through test problems for smooth and discontinuous solutions of the linear advection equation and the Euler equations of gas dynamics in one space dimension. Results are compared against exact solutions and against numerical results from well-known spatial reconstruction finite volume and DG schemes, with all methods implemented in the fully discrete ADER framework. The performance of GRPrec is very promising, especially in terms of efficiency, that is error against CPU cost.

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arXiv2025

Well-balanced high-order method for non-conservative hyperbolic PDEs with source terms: application to one-dimensional blood flow equations with gravity

The present work proposes a well-balanced finite volume-type numerical method for the solution of non-conservative hyperbolic partial differential equations (PDEs) with source terms. The method is characterized, first, by the use of a recently introduced high-order spatial reconstruction, based on generalized Riemann problem information from the previous time level. Such reconstruction is well-balanced up to order three, compact, efficient and easy to implement. Second, the method incorporates a well-balanced space-time evolution operator, which allows for well-balanced fully explicit time evolution. The accuracy and efficiency of the method are assessed on both a scalar problem (Burgers' equation) and a nonlinear PDE system (hyperbolized one-dimensional blood flow equations with gravity and friction, and with variable mechanical and geometrical properties). The well-balanced property is verified by showing that numerically-determined stationary solutions are preserved up to machine precision. The order of accuracy in space and time is validated through empirical convergence rate studies. Additionally, the performance of the method is assessed on a network of 86 arteries, under both stationary and transient conditions.

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arXiv2025

High-order fully well-balanced numerical methods for one-dimensional blood flow with discontinuous properties, friction and gravity

We present well-balanced, high-order, semi-discrete numerical schemes for one-dimensional blood flow models with discontinuous mechanical properties and algebraic source terms representing friction and gravity. While discontinuities in model parameters are handled using the Generalized Hydrostatic Reconstruction, the presence of algebraic source terms implies that steady state solutions cannot always be computed analytically. In fact, steady states are defined by an ordinary differential equation that needs to be integrated numerically. Therefore, we resort on a numerical reconstruction operator to identify and, where appropriate, preserve steady states with an accuracy that depends on the reconstruction operator's numerical scheme. We extend our methods to deal with networks of vessels and show numerical results for single- and multiple-vessel tests, including a network of 118 vessels, demonstrating the capacity of the presented methods to outperform naive discretizations of the equations under study.

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arXiv2022

Nonlinear lumped-parameter models for blood flow simulations in networks of vessels

To address the issue of computational efficiency related to the modelling of blood flow in complex networks, we derive a family of nonlinear lumped-parameter models for blood flow in compliant vessels departing from a well-established one-dimensional model. These 0D models must preserve important nonlinear properties of the original 1D model: the nonlinearity of the pressure-area relation and the pressure-dependent parameters characterizing the 0D models, the resistance $R$ and the inductance $L$, defined in terms of a time-dependent cross-sectional area subject to pressure changes. We introduce suitable coupling conditions to join 0D vessels through 0D junctions and construct 0D networks preserving the original 1D network topology. The newly derived nonlinear 0D models are then applied to several arterial networks and the predicted results are compared against (i) the reference 1D results, to validate the models and assess their ability to reproduce good approximations of pressure and flow waveforms in all vessels at a much lower computational cost, measured in terms of CPU time, and (ii) the linear 0D results, to evaluate the improvement gained by including certain nonlinearities in the 0D models, in terms of agreement with the 1D results.

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