In this report, we propose a collection of methods to make such an approach possible for Euler equations in one and two dimensions. We propose an explicit single-step ALE DG scheme for hyperbolic conservation laws. The scheme considerably reduces the numerical dissipations introduced by the Riemann solvers. We show that the scheme also preserves the constant states for any mesh motion. We then study the effect of mesh quality on the accuracy of the simulations, and based on that, come up with a mesh quality indicator for the ALE DG method. Based on the considerations from the study on mesh quality, we design a local mesh velocity algorithm to compute the motion of the mesh. And finally, we propose a local mesh adaptation algorithm to control the quality of the mesh, and prevent the mesh from degradation.
Nonlinear entropy stability analysis is used to derive entropy stable no-slip wall boundary conditions for the Eulerian model proposed by Svärd (Physica A: Statistical Mechanics and its Applications, 2018). and its spatial discretization based on entropy stable collocated discontinuous Galerkin operators with the summation-by-parts property for unstructured grids. A set of viscous test cases of increasing complexity are simulated using both the Eulerian and the classic compressible Navier-Stokes models. The numerical results obtained with the two models are compared, and differences and simila…
We present a class of Arbitrary Lagrangian-Eulerian hybridizable discontinuous Galerkin methods for the incompressible flow with moving boundaries and interfaces including two-phase flow with surface tension.
We present a Lagrangian-Eulerian scheme to solve the shallow water equations in the case of spatially variable bottom geometry. Using a local curvilinear reference system anchored on the bottom surface, we develop an effective first-order and high-resolution space-time discretization of the no-flow surfaces and solve a Lagrangian initial value problem that describes the evolution of the balance laws governing the geometrically intrinsic shallow water equations. The evolved solution set is then projected back to the original surface grid to complete the proposed Lagrangian-Eulerian formulation.…
We propose an explicit, single step discontinuous Galerkin (DG) method on moving grids using the arbitrary Lagrangian-Eulerian (ALE) approach for one dimensional Euler equations. The grid is moved with the local fluid velocity modified by some smoothing, which is found to considerably reduce the numerical dissipation introduced by Riemann solvers. The scheme preserves constant states for any mesh motion and we also study its positivity preservation property. Local grid refinement and coarsening are performed to maintain the mesh quality and avoid the appearance of very small or large cells. Se…
In this paper, we study both convergence and bounded variation properties of a new fully discrete conservative Lagrangian--Eulerian scheme to the entropy solution in the sense of Kruzhkov (scalar case) by using a weak asymptotic analysis. We discuss theoretical developments on the conception of no-flow curves for hyperbolic problems within scientific computing. The resulting algorithms have been proven to be effective to study nonlinear wave formations and rarefaction interactions. We present experiments to a study based on the use of the Wasserstein distance to show the effectiveness of the n…
An explicit high order semi-Lagrangian method is developed for solving Lagrangian transport equations in Eulerian-Lagrangian formulations. To ensure a semi-Lagrangian approximation that is consistent with an explicit Eulerian, discontinuous spectral element method (DSEM) discretization used for the Eulerian formulation, Lagrangian particles are seeded at Gauss quadrature collocation nodes within an element. The particles are integrated explicitly in time to obtain an advected polynomial solution at the advected Gauss quadrature locations. This approximation is mapped back in a semi-Lagrangian …
In this paper we present a novel technique for the simulation of moving boundaries and moving rigid bodies immersed in a rarefied gas using an Eulerian-Lagrangian formulation based on least square method. The rarefied gas is simulated by solving the Bhatnagar-Gross-Krook (BGK) model for the Boltzmann equation of rarefied gas dynamics. The BGK model is solved by an Arbitrary Lagrangian-Eulerian (ALE) method, where grid-points/particles are moved with the mean velocity of the gas. The computational domain for the rarefied gas changes with time due to the motion of the boundaries. To allow a simp…
We propose a new Eulerian-Lagrangian Runge-Kutta finite volume method for numerically solving convection and convection-diffusion equations. Eulerian-Lagrangian and semi-Lagrangian methods have grown in popularity mostly due to their ability to allow large time steps. Our proposed scheme is formulated by integrating the PDE on a space-time region partitioned by approximations of the characteristics determined from the Rankine-Hugoniot jump condition; and then rewriting the time-integral form into a time differential form to allow application of Runge-Kutta (RK) methods via the method-of-lines …
On Adaptive Eulerian-Lagrangian Method for Linear Convection-Diffusion Problems
The Pennsylvania State University（宾夕法尼亚州立大学）；Rutgers, The State University of New Jersey（新泽西州立罗格斯大学）；Institute of Computational Mathematics, Chinese Academy of Sciences（中国科学院计算数学与科学工程计算研究所）
We present a novel (high-order) hybridizable discontinuous Galerkin (HDG) scheme for the fluid-structure interaction (FSI) problem. The (moving domain) incompressible Navier-Stokes equations are discretized using a divergence-free HDG scheme within the arbitrary Lagrangian-Euler (ALE) framework. The nonlinear elasticity equations are discretized using a novel HDG scheme with an H(curl)-conforming velocity/displacement approximation. We further use a combination of the Nitsche's method (for the tangential component) and the mortar method (for the normal component) to enforce the interface condi…
We propose a generalized Eulerian-Lagrangian (GEL) discontinuous Galerkin (DG) method. The method is a generalization of the Eulerian-Lagrangian (EL) DG method for transport problems proposed in [arXiv preprint arXiv: 2002.02930 (2020)], which tracks solution along approximations to characteristics in the DG framework, allowing extra large time stepping size with stability. The newly proposed GEL DG method in this paper is motivated for solving linear hyperbolic systems with variable coefficients, where the velocity field for adjoint problems of the test functions is frozen to constant. In thi…
This article presents a novel and practically useful link between geometric integration, low-discrepancy sampling and code coupling for Lagrangian and Eulerian Vlasov-Poisson solvers. Low-discrepancy sequences, also called quasi-random sequences (Quasi Monte Carlo), provide convergence rates close to $\mathcal{O}( N^{-1})$ which are far superior to (pseudo) random numbers (Monte Carlo) settling in at only $\mathcal{O}(N^{-0.5})$. Lagrangian particle methods such as PIC rely on Monte Carlo integration. The particle distributions are nonlinearly perturbed by the forward flow following the charac…
The immersed boundary (IB) method is a non-body conforming approach to fluid-structure interaction (FSI) that uses an Eulerian description of the momentum, viscosity, and incompressibility of a coupled fluid-structure system and a Lagrangian description of the deformations, stresses, and resultant forces of the immersed structure. Integral transforms with Dirac delta function kernels couple the Eulerian and Lagrangian variables, and in practice, discretizations of these integral transforms use regularized delta function kernels. Many different kernel functions have been proposed, but prior num…
This paper develops the structure-preserving finite volume weighted essentially non-oscillatory (WENO) hybrid schemes for the shallow water equations under the arbitrary Lagrangian-Eulerian (ALE) framework, dubbed as ALE-WENO schemes. The WENO hybrid reconstruction is adopted on moving meshes, which distinguishes the smooth, non-smooth, and transition stencils by a simple smoothness detector. To maintain the positivity preserving and the well-balanced properties of the ALE-WENO schemes, we adapt the positivity preserving limiter and the well-balanced approaches on static meshes to moving meshe…
In this paper, we construct a novel Eulerian-Lagrangian finite volume (ELFV) method for nonlinear scalar hyperbolic equations in one space dimension. It is well known that the exact solutions to such problems may contain shocks though the initial conditions are smooth, and direct numerical methods may suffer from restricted time step sizes. To relieve the restriction, we propose an ELFV method, where the space-time domain was separated by the partition lines originated from the cell interfaces whose slopes are obtained following the Rakine-Hugoniot junmp condition. Unfortunately, to avoid the …
In this paper we introduce a numerical scheme for fluid-structure interaction problems in two or three space dimensions: A flexible elastic plate is interacting with a viscous, compressible barotropic fluid. Hence the physical domain of definition (the domain of Eulerian coordinates) is changing in time. We introduce a fully discrete scheme that is stable, satisfies geometric conservation, mass conservation and the positivity of the density. We also prove that the scheme is consistent with the definition of continuous weak solutions.
Hyperbolic partial differential equations (PDEs) cover a wide range of interesting phenomena, from human and hearth-sciences up to astrophysics: this unavoidably requires the treatment of many space and time scales in order to describe at the same time observer-size macrostructures, multi-scale turbulent features, and also zero-scale shocks. Moreover, numerical methods for solving hyperbolic PDEs must reliably handle different families of waves: smooth rarefactions, and discontinuities of shock and contact type. In order to achieve these goals, an effective approach consists in the combination…
In this paper we employ two implementations of the fictitious domain (FD) method to simulate water-entry and water-exit problems and demonstrate their ability to simulate practical marine engineering problems. In FD methods, the fluid momentum equation is extended within the solid domain using an additional body force that constrains the structure velocity to be that of a rigid body. Using this formulation, a single set of equations is solved over the entire computational domain. The constraint force is calculated in two distinct ways: one using an Eulerian-Lagrangian framework of the immersed…
Motivated by challenges in Earth mantle convection, we present a massively parallel implementation of an Eulerian-Lagrangian method for the advection-diffusion equation in the advection-dominated regime. The advection term is treated by a particle-based, characteristics method coupled to a block-structured finite-element framework. Its numerical and computational performance is evaluated in multiple, two- and three-dimensional benchmarks, including curved geometries, discontinuous solutions, pure advection, and it is applied to a coupled non-linear system modeling buoyancy-driven convection in…
We propose an Eulerian-Lagrangian (EL) Runge-Kutta (RK) discontinuous Galerkin (DG) method for wave equations. The method is designed based on the ELDG method for transport problems [J. Comput. Phy. 446: 110632, 2021.], which tracks solution along approximations to characteristics in the DG framework, allowing extra large time stepping sizes with stability. The wave equation can be written as a first order hyperbolic system. Considering each characteristic family, a straightforward application of ELDG will be to transform to the characteristic variables, evolve them on associated characteristi…
We present a novel second-order semi-implicit hybrid finite volume / finite element (FV/FE) scheme for the numerical solution of the incompressible and weakly compressible Navier-Stokes equations on moving unstructured meshes using an Arbitrary-Lagrangian-Eulerian (ALE) formulation. The scheme is based on a suitable splitting of the governing PDE into subsystems and employs staggered grids, where the pressure is defined on the primal simplex mesh, while the velocity and the remaining flow quantities are defined on an edge-based staggered dual mesh. The key idea of the scheme is to discretize t…
DNS of compressible multiphase flows through the Eulerian approach
Scuola Normale Superiore di Pisa（比萨高等师范学院）；Istituto Nazionale di Vulcanologia e Geofisica（意大利火山学与地球物理研究所）；Istituto Nazionale di Geofisica e Vulcanologia（意大利地球物理与火山学研究所）
Direct numerical simulation of a compressible multiphase flow through the fast Eulerian approach
Scuola Normale Superiore di Pisa（比萨高等师范学院）；Istituto Nazionale di Vulcanologia e Geofisica（火山学与地球物理学国家研究所）；Istituto Nazionale di Geofisica e Vulcanologia（地球物理学与火山学国家研究所）
We propose a new Eulerian-Lagrangian (EL) discontinuous Galerkin (DG) method. The method is designed as a generalization of the semi-Lagrangian (SL) DG method for linear advection problems proposed in [J. Sci. Comput. 73: 514-542, 2017], which is formulated based on an adjoint problem and tracing upstream cells by tracking characteristics curves highly accurately. In the SLDG method, depending on the velocity field, upstream cells could be of arbitrary shape. Thus, a more sophisticated approximation to sides of the upstream cells is required to get high order approximation. For example, quadra…
ASHEE: a compressible, equilibrium-Eulerian model for volcanic ash plumes
Scuola Normale Superiore, Pisa, Italy（比萨高等师范学院）；Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Pisa, Italy（意大利国家地球物理与火山学研究所）；Dipartimento di Matematica, Università degli Studi di Pisa, Italy（比萨大学数学系）
We present a novel framework based on semi-bounded spatial operators for analyzing and discretizing initial boundary value problems on moving and deforming domains. This development extends an existing framework for well-posed problems and energy stable discretizations from stationary domains to the general case including arbitrary mesh motion. In particular, we show that an energy estimate derived in the physical coordinate system is equivalent to a semi-bounded property with respect to a stationary reference domain. The continuous analysis leading up to this result is based on a skew-symmetr…
