In the present work, a high order finite element type residual distribution scheme is designed in the framework of multidimensional compressible Euler equations of gas dynamics. The strengths of the proposed approximation rely on the generic spatial discretization of the model equations using a continuous finite element type approximation technique, while avoiding the solution of a large linear system with a sparse mass matrix which would come along with any standard ODE solver in a classical finite element approach to advance the solution in time. In this work, we propose a new Residual Distribution (RD) scheme, which provides an arbitrary explicit high order approximation of the smooth solutions of the Euler equations both in space and time. The design of the scheme via the coupling of the RD formulation [1, 2] with a Deferred Correction (DeC) type method [3,4], allows to have the matrix associated to the update in time, which needs to be inverted, to be diagonal. The use of Bernstein polynomials as shape functions, guarantees that this diagonal matrix is invertible and ensures strict positivity of the resulting diagonal matrix coefficients. This work is the extension of [5, 6] to multidimensional systems. We have assessed our method on several challenging benchmark problems for one-and two-dimensional Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions. polynomials are a suitable choice, but this is not the only possible one. The idea to use as shape functions the Bernstein polynomials, instead of the more typical Lagrange polynomials, has been discussed in [7,6] applied to the context of high order residual distribution schemes and very recently, in [8], this idea has been applied for a different class of methods, namely, the flux-corrected transport method.The purpose of this paper is to show how these ideas can be further extended for solving the Euler equations of fluid dynamics for the simulation of flows involving strong discontinuities. The RD formulation used here is based on the finite element approximation of the solution as a globally continuous piecewise polynomial. The design principle of the new RD scheme guarantees a compact approximation stencil even for high order accuracy, which would hold for Discontinuous Galerkin (DG), but not for example for Finite Volume (FV) methods and allows to consider a smaller number of nodes than DG ([9, 10, 11]).The format of this paper is the following. In Section 2, we recall the idea of the residual distribution schemes for steady problems and in Section 3 we describe the time-stepping algorithm and adapt the method developed in [5,6] to multidimensional systems. We illustrate the robustness and accuracy of the proposed method by means of rigorous numerical tests and discuss the obtained results in Section 4. Finally, we give the conclusive remarks and outline further perspectives.
It is well known, thanks to Lax-Wendroff theorem, that the local conservation of a numerical scheme for a conservative hyperbolic system is a simple and systematic way to guarantee that, if stable, a scheme will provide a sequence of solutions that will converge to a weak solution of the continuous problem. In [1], it is shown that a nonconservative scheme will not provide a good solution. The question of using, nevertheless, a nonconservative formulation of the system and getting the correct solution has been a long-standing debate. In this paper, we show how to get a relevant weak solution from a pressure-based formulation of the Euler equations of fluid mechanics. This is useful when dealing with nonlinear equations of state because it is easier to compute the internal energy from the pressure than the opposite. This makes it possible to get oscillation-free solutions, contrarily to classical conservative methods. An extension to multiphase flows is also discussed.
The aim of the present work is to develop a model able to represent the propagation and transformation of waves in nearshore areas. The focus is on the phenomena of wave breaking, shoaling and run-up. These different phenomena are represented through a hybrid approach obtained by the coupling of non-linear Shallow Water equations with the extended Boussinesq equations of Madsen and Sørensen. The novelty is the switch tool between the two modelling equations: a critical free surface Froude criterion. This is based on a physically meaningful new approach to detect wave breaking, which corresponds to the steepening of the wave's crest which turns into a roller. To allow for an appropriate discretization of both types of equations, we consider a finite element Upwind Petrov Galerkin method with a novel limiting strategy, that guarantees the preservation of smooth waves as well as the monotonicity of the results in presence of discontinuities. We provide a detailed discussion of the implementation of the newly proposed detection method, as well as of two other well known criteria which are used for comparison. An extensive benchmarking on several problems involving different wave phenomena and breaking conditions allows to show the robustness of the numerical method proposed, as well as to assess the advantages and limitations of the different detection methods.Implementation and Evaluation of Breaking Detection Criteria for a Hybrid Boussinesq Model are defined via physical arguments, or through some auxiliary evolution model (see e.g.[24] and references therein), and, eventually, adjusted by means of numerical experiments. An alternative to the eddy viscosity type modeling are the roller approaches, that account more explicitly for the large scale effects of vorticity on the mean flow. Finally, the hybrid approaches exploit the properties of hyperbolic conservation laws endowed with an entropy, and model large scale effects of wave breaking with the dissipation of the total energy across shocks arising in shallow water simulations or augment/modify Boussinesq-type equations to include, for example, additional terms caused by the presence of the surface rollers (see cf. [41]). Independently of the chosen closure strategy, some breaking detection criteria is very often required to trigger the onset of wave breaking. In practice it is this criteria that leads to the activation of the closure. As well explained in [33], historically the first detection criteria were based on quantities computed using information over one full phase of the wave. The possibility of performing accurate phase resolved simulations, has led to the necessity of formulating new detection criteria based on local flow features [38,25,42,23]. More specifically, these criteria rely on a wave-by-wave analysis more efficient to program in the context of phase resolved simulations, and providing a physically correct detection of breaking onset and termination, provided that a sufficiently accurate description of the wave profiles is available. In this...
We are interested in the numerical solution of linear hyperbolic problems using continuous finite elements of arbitrary order. It is well known that this kind of methods, once the weak formulation has been written, leads to a system of ordinary differential equations in R N , where N is the number of degrees of freedom. The solution of the resulting ODE system involves the inversion of a sparse mass matrix that is not block diagonal. Here we show how to avoid this step, and what are the consequences of the choice of the finite element space. Numerical examples show the correctness of our approach.
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