90 research outputs found
An all speed second order well-balanced IMEX relaxation scheme for the Euler equations with gravity
We present an implicit-explicit well-balanced finite volume scheme for the Euler equations with a gravitational source term which is able to deal also with low Mach flows. To visualize the different scales we use the non-dimensionalized equations on which we apply a pressure splitting and a Suliciu relaxation. On the resulting model, we apply a splitting of the flux into a linear implicit and an non-linear explicit part that leads to a scale independent time-step. The explicit step consists of a Godunov type method based on an approximative Riemann solver where the source term is included in the flux formulation. We develop the method for a first order scheme and give an extension to second order. Both schemes are designed to be well-balanced, preserve the positivity of density and internal energy and have a scale independent diffusion. We give the low Mach limit equations for well-prepared data and show that the scheme is asymptotic preserving. These properties are numerically validated by various test cases
Numerical methods for all-speed flows for the Euler equations including well-balancing of source terms
This thesis regards the numerical simulation of inviscid compressible ideal gases which are described
by the Euler equations. We propose a novel implicit explicit (IMEX) relaxation scheme to simulate
flows from compressible as well as near incompressible regimes based on a Suliciu-type relaxation
model. The Mach number plays an important role in the design of the scheme, as it has great
influence on the flow behaviour and physical properties of solutions of the Euler equations. Our
focus is on an accurate resolution of the Mach number independent material wave. A special feature
of our scheme is that it can account for the influence of a gravitational field on the fluid flow and is
applicable also in small Froude number regimes. The time step of the IMEX scheme is constrained
only by the eigenvalues of the explicitly treated part and is independent of the Mach number allowing
for large time steps independent of the flow regime. In addition, the scheme is provably asymptotic
preserving and well-balanced for arbitrary a priori known hydrostatic equilibria independently of
the considered Mach and Froude regime. Also, the scheme preserves the positivity of density and
internal energy throughout the simulation, it is well suited for physical applications. To increase the
accuracy, a natural extension to second order is provided. The theoretical properties of the given
schemes are numerically validated by various test cases performed on Cartesian grids in multiple
space dimensions
A well-balanced semi-implicit IMEX finite volume scheme for ideal Magnetohydrodynamics at all Mach numbers
We propose a second-order accurate semi-implicit and well-balanced finite
volume scheme for the equations of ideal magnetohydrodynamics (MHD) including
gravitational source terms. The scheme treats all terms associated with the
acoustic pressure implicitly while keeping the remaining terms part of the
explicit sub-system. This semi-implicit approach makes the method particularly
well suited for problems in the low Mach regime. We combine the semi-implicit
scheme with the deviation well-balancing technique and prove that it maintains
equilibrium solutions for the magnetohydrostatic case up to rounding errors. In
order to preserve the divergence-free property of the magnetic field enforced
by the solenoidal constraint, we incorporate a constrained transport method in
the semi-implicit framework. Second order of accuracy is achieved by means of a
standard spatial reconstruction technique with total variation diminishing
(TVD) property, and by an asymptotic preserving (AP) time stepping algorithm
built upon the implicit-explicit (IMEX) Runge-Kutta time integrators. Numerical
tests in the low Mach regime and near magnetohydrostatic equilibria support the
low Mach and well-balanced properties of the numerical method
Implicit and semi-implicit well-balanced finite-volume methods for systems of balance laws
The aim of this work is to design implicit and semi-implicit high-order well-balanced finite-volume numerical methods for 1D systems of balance laws. The strategy introduced by two of the authors in some previous papers for explicit schemes based on the application of a well-balanced reconstruction operator is applied. The well-balanced property is preserved when quadrature formulas are used to approximate the averages and the integral of the source term in the cells. Concerning the time evolution, this technique is combined with a time discretization method of type RK-IMEX or RK-implicit. The methodology will be applied to several systems of balance laws.This work is partially supported by projects RTI2018-096064-B-C21 funded by MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe”, projects P18-RT-3163 of Junta de AndalucĂa and UMA18-FEDERJA-161 of Junta de AndalucĂa-FEDER-University of Málaga. G.Russo and S.Boscarino acknowledge partial support from the Italian Ministry of University and Research (MIUR), PRIN Project 2017 (No. 2017KKJP4X) entitled “Innovative numerical methods for evolu-tionary partial differential equations and applications”. I. GĂłmez-Bueno is also supported by a Grant from “El Ministerio de Ciencia, InnovaciĂłn y Universidades”, Spain (FPU2019/01541) funded by MCIN/AEI/10.13039/501100011033 and “ESF Invest-ing in your future”. // Funding for open access charge: Universidad de Málaga/CBUA
A large time-step and well-balanced Lagrange-Projection type scheme for the shallow-water equations
This work focuses on the numerical approximation of the Shallow Water
Equations (SWE) using a Lagrange-Projection type approach. We propose to extend
to this context recent implicit-explicit schemes developed in the framework of
compressibleflows, with or without stiff source terms. These methods enable the
use of time steps that are no longer constrained by the sound velocity thanks
to an implicit treatment of the acoustic waves, and maintain accuracy in the
subsonic regime thanks to an explicit treatment of the material waves. In the
present setting, a particular attention will be also given to the
discretization of the non-conservative terms in SWE and more specifically to
the well-known well-balanced property. We prove that the proposed numerical
strategy enjoys important non linear stability properties and we illustrate its
behaviour past several relevant test cases
AnAll Speed SecondOrder IMEXRelaxation Scheme for the Euler Equations
We present an implicit-explicit finite volume scheme for the Euler equations. We start from the non-dimensionalised Euler equations where we split the pressure in a slow and a fast acoustic part. We use a Suliciu type relaxation model which we split in an explicit part, solved using a Godunov-type scheme based on an approximate Riemann solver, and an implicit part where we solve an elliptic equation for the fast pressure. The relaxation source terms are treated projecting the solution on the equilibrium manifold. The proposed scheme is positivity preserving with respect to the density and internal energy and asymptotic preserving towards the incompressible Euler equations. For this first order scheme we give a second order extension which maintains the positivity property. We perform numerical experiments in 1D and 2D to show the applicability of the proposed splitting and give convergence results for the second order extension
A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers
We present a novel staggered semi-implicit hybrid FV/FE method for the
numerical solution of the shallow water equations at all Froude numbers on
unstructured meshes. A semi-discretization in time of the conservative
Saint-Venant equations with bottom friction terms leads to its decomposition
into a first order hyperbolic subsystem containing the nonlinear convective
term and a second order wave equation for the pressure. For the spatial
discretization of the free surface elevation an unstructured mesh of triangular
simplex elements is considered, whereas a dual grid of the edge-type is
employed for the computation of the depth-averaged momentum vector. The first
stage of the proposed algorithm consists in the solution of the nonlinear
convective subsystem using an explicit Godunov-type FV method on the staggered
grid. Next, a classical continuous FE scheme provides the free surface
elevation at the vertex of the primal mesh. The semi-implicit strategy followed
circumvents the contribution of the surface wave celerity to the CFL-type time
step restriction making the proposed algorithm well-suited for low Froude
number flows. The conservative formulation of the governing equations also
allows the discretization of high Froude number flows with shock waves. As
such, the new hybrid FV/FE scheme is able to deal simultaneously with both,
subcritical as well as supercritical flows. Besides, the algorithm is well
balanced by construction. The accuracy of the overall methodology is studied
numerically and the C-property is proven theoretically and validated via
numerical experiments. The solution of several Riemann problems attests the
robustness of the new method to deal also with flows containing bores and
discontinuities. Finally, a 3D dam break problem over a dry bottom is studied
and our numerical results are successfully compared with numerical reference
solutions and experimental data
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