This paper presents an asymptotic preserving (AP) all Mach number finite volume shock capturing method for the numerical solution of compressible Euler equations of gas dynamics. Both isentropic and full Euler equations are considered. The equations are discretized on a staggered grid. This simplifies flux computation and guarantees a natural central discretization in the low Mach limit, thus dramatically reducing the excessive numerical diffusion of upwind discretizations. Furthermore, second order accuracy in space is automatically guaranteed. For the time discretization we adopt an Semi-IMplicit/EXplicit (S-IMEX) discretization getting an elliptic equation for the pressure in the isentropic case and for the energy in the full Euler equations. Such equations can be solved linearly so that we do not need any iterative solver thus reducing computational cost. Second order in time is obtained by a suitable S-IMEX strategy taken from Boscarino et al. in [6]. Moreover, the CFL stability condition is independent of the Mach number and depends essentially on the fluid velocity. Numerical tests are displayed in one and two dimensions to demonstrate performances of our scheme in both compressible and incompressible regimes.

We study numerical methods that are inspired by the active flux method of Eymann and Roe and present several new results for one and two-dimensional hyperbolic problems. For one-dimensional linear problems we show that the unlimited active flux method can be interpreted as an ADER method. This interpretation motivates the construction of new third order accurate methods for nonlinear hyperbolic conservation laws. In the two-dimensional case, equivalent methods are only obtained for scalar linear problems. For two-dimensional linear systems the methods are no longer equivalent. For the two-dimensional acoustic equations we compare the accuracy of the two resulting approaches. While commonly used methods for hyperbolic problems are based on discontinuous reconstructions, the active flux method uses a continuous, piecewise quadratic reconstruction. For nonlinear problems we identify a situation in which the continuous reconstruction leads to an unstable approximation. We propose a limiting strategy which overcomes this problem. Our limited version of the active flux method uses the same local stencil as the original method. Keywords Finite volume methods • Hyperbolic conservation laws • Active flux method • ADER method • High-order methods Mathematics Subject Classification 65M08 • 65M25 Recently, Eymann, Roe and coauthors [2-4,12,14] introduced a new numerical method for hyperbolic conservation laws, which they called the active flux method. For sufficiently smooth linear problems the method is third order accurate [4]. In [12], third order accurate This work was supported by the DFG through HE 4858/4-1.

Reduced order modeling (ROM) techniques are numerical methods that approximate the solution of parametric partial differential equation (PED) by properly combining the high-fidelity solutions of the problem obtained for several configurations, i.e. for several properly chosen values of the physical/geometrical parameters characterizing the problem. By starting from a database of high-fidelity solutions related to a certain values of the parameters, we apply the proper orthogonal decomposition with interpolation (PODI) and then reconstruct the variables of interest for new values of the parameters, i.e. different values from the ones included in the database. Furthermore, we present a preliminary web application through which one can run the ROM with a very user-friendly approach, without the need of having expertise in the numerical analysis and scientific computing field. The case study we have chosen to test the efficiency of our algorithm is represented by the aortic blood flow pattern in presence of a left ventricular (LVAD) assist device when varying the pump flow rate.

Reduced order modeling (ROM) techniques are numerical methods that approximate the solution to parametric partial differential equation (PDE) is approximated by properly combining the high-fidelity solutions of the problem obtained for several configurations, i.e. for several properly chosen values of the physical/geometrical parameters characterizing the problem. In this contribution, we propose an efficient non-intrusive data-driven framework involving ROM techniques in computational fluid dynamics (CFD) for hemodynamics applications. By starting from a database of high-fidelity solutions related to a certain values of the parameters, we apply the proper orthogonal decomposition with interpolation (PODI) and then reconstruct the variables of interest for new values of the parameters, i.e. different values from the ones included in the database. Furthermore, we present a preliminary web application through which one can run the ROM with a very user-friendly approach, without the need of having expertise in the numerical analysis and scientific computing field. The case study we have chosen to test the efficiency of our algorithm is represented by the aortic blood flow pattern in presence of a Left Ventricular Assist Device (LVAD) when varying the pump flow rate.

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