The shallow water equations in Lagrangian coordinates

Recent advances in the collection of Lagrangian data from the ocean and results about the well-posedness of the primitive equations have led to a renewed interest in solving flow equations in Lagrangian coordinates. We do not take the view that solving in Lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted. Rather, the grid in Lagrangian coordinates represents the initial position of particles, and it does not change with time. We apply numerical methods traditionally used to solve differential equations in Eulerian coordinates, to solve the shallow water equations in Lagrangian coordinates. The difficulty with solving in Lagrangian coordinates is that the transformation from Eulerian coordinates results in solving a highly nonlinear partial differential equation. The non-linearity is mainly due to the Jacobian of the coordinate transformation, which is a precise record of how the particles are rotated and stretched. The inverse Jacobian must be calculated, thus Lagrangian coordinates cannot be used in instances where the Jacobian vanishes. For linear (spatial) flows we give an explicit formula for the Jacobian and describe the two situations where the Lagrangian shallow water equations cannot be used because either the Jacobian vanishes or the shallow water assumption is violated. We also prove that linear (in space) steady state solutions of the Lagrangian shallow water equations have Jacobian equal to one. In the situations where the shallow water equations can be solved in Lagrangian coordinates, accurate numerical solutions are found with finite differences, the Chebyshev pseudospectral method, and the fourth order Runge–Kutta method. The numerical results shown here emphasize the need for high order temporal approximations for long time integrations.     

An immersed boundary energy-based method for incompressible viscoelasticity

Incompressible viscoelastic materials are prevalent in biological applications. In this paper we present a method for incompressible viscoelasticity in which the elasticity of the material is described in Lagrangian form (i.e. in material coordinates), and Eulerian (spatial) coordinates are used for the equations of motion and to enforce the incompressibility condition. The elastic forces are computed directly from an energy functional without the use of stress tensors, and the immersed boundary method is used to communicate between Lagrangian and Eulerian variables. The method is first applied to a warm-up problem, in which a viscoelastic incompressible material fills a two-dimensional periodic domain. For this problem, we study convergence of the velocity field, the deformation map, and the Eulerian force density. The numerical results indicate that the velocity field and deformation map converge strongly at second order and the Eulerian force density converges weakly at second order. Incompressibility is well maintained, as indicated by area conservation in this 2D problem. Finally, the method is applied to a three-dimensional fluid–structure interaction problem with two different materials: an isotropic neo-Hookean model and an anisotropic fiber-reinforced model.     

A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations

The Vlasov–Poisson equations describe the evolution of a collisionless plasma, represented through a probability density function (PDF) that self-interacts via an electrostatic force. One of the main difficulties in numerically solving this system is the severe time-step restriction that arises from parts of the PDF associated with moderate-to-large velocities. The dominant approach in the plasma physics community for removing these time-step restrictions is the so-called particle-in-cell (PIC) method, which discretizes the distribution function into a set of macro-particles, while the electric field is represented on a mesh. Several alternatives to this approach exist, including fully Lagrangian, fully Eulerian, and so-called semi-Lagrangian methods. The focus of this work is the semi-Lagrangian approach, which begins with a grid-based Eulerian representation of both the PDF and the electric field, then evolves the PDF via Lagrangian dynamics, and finally projects this evolved field back onto the original Eulerian mesh. In particular, we develop in this work a method that discretizes the 1 + 1 Vlasov–Poisson system via a high-order discontinuous Galerkin (DG) method in phase space, and an operator split, semi-Lagrangian method in time. Second-order accuracy in time is relatively easy to achieve via Strang operator splitting. With additional work, using higher-order splitting and a higher-order method of characteristics, we also demonstrate how to push this scheme to fourth-order accuracy in time. We show how to resolve all of the Lagrangian dynamics in such a way that mass is exactly conserved, positivity is maintained, and high-order accuracy is achieved. The Poisson equation is solved to high-order via the smallest stencil local discontinuous Galerkin (LDG) approach. We test the proposed scheme on several standard test cases.     

A new high-order discontinuous Galerkin spectral finite element method for Lagrangian gas dynamics in two-dimensions

This paper presents a new high-order cell-centered Lagrangian scheme for two-dimensional compressible flow. The scheme uses a fully Lagrangian form of the gas dynamics equations, which is a weakly hyperbolic system of conservation laws. The system of equations is discretized in the Lagrangian space by discontinuous Galerkin method using a spectral basis. The vertex velocities and the numerical fluxes through the cell interfaces are computed consistently in the Eulerian space by virtue of an improved nodal solver. The nodal solver uses the HLLC approximate Riemann solver to compute the velocities of the vertex. The time marching is implemented by a class of TVD Runge–Kutta type methods. A new HWENO (Hermite WENO) reconstruction algorithm is developed and used as limiters for RKDG methods to maintain compactness of RKDG methods. The scheme is conservative for the mass, momentum and total energy. It can maintain high-order accuracy both in space and time, obey the geometrical conservation law, and achieve at least second order accuracy on quadrilateral meshes. Results of some numerical tests are presented to demonstrate the accuracy and the robustness of the scheme.     

Vorticity–divergence mass-conserving semi-Lagrangian shallow-water model using the reduced grid on the sphere

The semi-Lagrangian semi-implicit shallow water model on the sphere using the reduced latitude–longitude grid is presented. The key feature of the model is the vorticity–divergence formulation on the unstaggered grid. The new algorithm for the reconstruction of wind components from vorticity and divergence is described. The mass-conservative version of the model is developed. The conservative cascade scheme (CCS) by Nair et al. is modified to provide a locally-conservative semi-Lagrangian advection algorithm for the reduced grid. Some numerical advection tests are carried out to demonstrate the accuracy of the CCS with the reduced grid. The CCS-based discretization for the continuity equation and finite-volume Helmholtz problem solver are introduced to guarantee the mass-conservation.The results for shallow water tests on the sphere are presented. The results for different versions of the model are compared. They are also compared with the results for the same tests available in literature. The impact of the reduced grid is analyzed. The mass-conservative version of the model using the reduced grid with up to 20% reduction of grid points number has approximately the same accuracy as its non-conservative counterpart implemented on the regular latitude–longitude grid.     

Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow

In this paper, we propose a semi-Lagrangian finite difference formulation for approximating conservative form of advection equations with general variable coefficients. Compared with the traditional semi-Lagrangian finite difference schemes [5], [25], which approximate the advective form of the equation via direct characteristics tracing, the scheme proposed in this paper approximates the conservative form of the equation. This essential difference makes the proposed scheme naturally conservative for equations with general variable coefficients. The proposed conservative semi-Lagrangian finite difference framework is coupled with high order essentially non-oscillatory (ENO) or weighted ENO (WENO) reconstructions to achieve high order accuracy in smooth parts of the solution and to capture sharp interfaces without introducing spurious oscillations. The scheme is extended to high dimensional problems by Strang splitting. The performance of the proposed schemes is demonstrated by linear advection, rigid body rotation, swirling deformation, and two dimensional incompressible flow simulation in the vorticity stream-function formulation. As the information is propagating along characteristics, the proposed scheme does not have the CFL time step restriction of the Eulerian method, allowing for a more efficient numerical realization for many application problems.     

An arbitrary Lagrangian–Eulerian formulation for the numerical simulation of flow patterns generated by the hydromedusa Aequorea victoria

A new geometrically conservative arbitrary Lagrangian–Eulerian (ALE) formulation is presented for the moving boundary problems in the swirl-free cylindrical coordinates. The governing equations are multiplied with the radial distance and integrated over arbitrary moving Lagrangian–Eulerian quadrilateral elements. Therefore, the continuity and the geometric conservation equations take very simple form similar to those of the Cartesian coordinates. The continuity equation is satisfied exactly within each element and a special attention is given to satisfy the geometric conservation law (GCL) at the discrete level. The equation of motion of a deforming body is solved in addition to the Navier–Stokes equations in a fully-coupled form. The mesh deformation is achieved by solving the linear elasticity equation at each time level while avoiding remeshing in order to enhance numerical robustness. The resulting algebraic linear systems are solved using an ILU(k) preconditioned GMRES method provided by the PETSc library. The present ALE method is validated for the steady and oscillatory flow around a sphere in a cylindrical tube and applied to the investigation of the flow patterns around a free-swimming hydromedusa Aequorea victoria (crystal jellyfish). The calculations for the hydromedusa indicate the shed of the opposite signed vortex rings very close to each other and the formation of large induced velocities along the line of interaction while the ring vortices moving away from the hydromedusa. In addition, the propulsion efficiency of the free-swimming hydromedusa is computed and its value is compared with values from the literature for several other species.     

Accurate interface-tracking of surfaces in three dimensions for arbitrary Lagrangian–Eulerian schemes

We extend the computational method presented in [1] for tracking an interface immersed in a given velocity field to three spatial dimensions. The proposed method is particularly relevant to the simulation of unsteady free surface problems using the arbitrary Lagrangian–Eulerian framework, and has been constructed with two goals in mind: (i) to be able to accurately follow the interface; and (ii) to automatically maintain a good distribution of the grid points along the interface. The method combines information from a pure Lagrangian approach with information from an ALE approach. The new method offers flexibility in terms of how an “optimal” point distribution should be defined, and relies on the solution of two-dimensional surface convection problems. We verify the new method by solving model problems both in the single and multiple spectral element case, and we compare this method with other traditional alternatives. We have been able to verify first, second, and third order temporal accuracy for the new method by solving these three-dimensional model problems.     

Eulerian–Lagrangian multiscale methods for solving scalar equations – Application to incompressible two-phase flows

The present article proposes a new hybrid Eulerian–Lagrangian numerical method, based on a volume particle meshing of the Eulerian grid, for solving transport equations. The approach, called Volume Of Fluid Sub-Mesh method (VOF-SM), has the advantage of being able to deal with interface tracking as well as advection–diffusion transport equations of scalar quantities. The Eulerian evolutions of a scalar field could be obtained on any orthogonal curvilinear grid thanks to the Lagrangian advection and a redistribution of particles on the Eulerian grid. The Eulerian concentrations result from the projection of the volume and scalar informations handled by the particles. The particle velocities are interpolated from the Eulerian velocity field. The VOF-SM method is validated on several scalar interface tracking and transport problems and is compared to existing schemes within the literature. It is finally coupled to a Navier–Stokes solver and applied to the simulation of two free-surface flows, i.e. the two-dimensional buckling of a viscous jet during the filling of a square mold and the three-dimensional dam-break flow in a tank.     

Numerical visualization of free surface oscillation predicted with arbitrary Lagrangian-Eulerian method

A numerical prediction method has been proposed to predict non-linear free surface oscillation in a three-dimensional container. The fluid motions are numerically predicted with Navier-Stokes equations discretized in a Lagrangian scheme with sufficient numerical accuracy. The profile of a free surface is precisely represented with three-dimensional body-fitted coordinates (BFC), which are regenerated in each computational step on the basis of the arbitrary Lagrangian-Eulerian (ALE) formulation. The computational method was applied to non-linear sloshings and transitions from sloshing to swirling motions. The predicted free surface motions were visualized as sequential image files and animations to understand their dynamic futures     

A cell-centered Lagrangian scheme with the preservation of symmetry and conservation properties for compressible fluid flows in two-dimensional cylindrical geometry

We develop a new cell-centered control volume Lagrangian scheme for solving Euler equations of compressible gas dynamics in cylindrical coordinates. The scheme is designed to be able to preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid. Unlike many previous area-weighted schemes that possess the spherical symmetry property, our scheme is discretized on the true volume and it can preserve the conservation property for all the conserved variables including density, momentum and total energy. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry, accuracy and non-oscillatory properties.     

Numerical simulations of instabilities in the implosion process of inertial confined fusion in 2D cylindrical coordinates

In this paper, we introduce a multi-material arbitrary Lagrangian and Eulerian method for the hydrodynamic radiative multi-group diffusion model in 2D cylindrical coordinates. The basic idea in the construction of the method is the following: In the Lagrangian step, a closure model of radiation-hydrodynamics is used to give the states of equations for materials in mixed cells. In the mesh rezoning step, we couple the rezoning principle with the Lagrangian interface tracking method and an Eulerian interface capturing scheme to compute interfaces sharply according to their deformation and to keep cells in good geometric quality. In the interface reconstruction step, a dual-material Moment-of-Fluid method is introduced to obtain the unique interface in mixed cells. In the remapping step, a conservative remapping algorithm of conserved quantities is presented. A number of numerical tests are carried out and the numerical results show that the new method can simulate instabilities in complex fluid field under large deformation,and are accurate and robust.     

基于CE/SE方法的二维Euler型多物质流体弹塑性问题计算

将CE/SE方法推广到二维固体流体弹塑性问题的数值计算,同时结合杂交粒子水平集方法追踪物质界面和合适的边界条件,提出一套完整的二维Euler型流体弹塑性计算方案.通过长钨杆侵彻装甲钢实验的数值模拟,对方法的精度和有效性进行验证.     

ELLAM for resolving the kinematics of two-dimensional resistive magnetohydrodynamic flows

We combine the finite element method with the Eulerian–Lagrangian Localized Adjoint Method (ELLAM) to solve the convection–diffusion equations that describe the kinematics of magnetohydrodynamic flows, i.e., the advection and diffusion of a magnetic field. Simulations of three two-dimensional test problems are presented and in each case we analyze the energy of the magnetic field as it evolves towards its equilibrium state. Our numerical results highlight the accuracy and efficiency of the ELLAM approach for convection-dominated problems.     

基于近似Riemann解的有限体积ALE方法

研究二维平面坐标系和二维轴对称坐标系中四边形网格上可压缩流体力学的有限体积ALE(Arbitrary Lagrangian Eulerian)方法.数值方法采用节点中心有限体积法,数值通量采用适用于任意状态方程的HLLC(Harten-Lax-Van Leer-Collela)通量.空间二阶精度通过用WENO(weighted essentially non-oscillatory)方法对原始变量进行重构获得,时间离散采用两步显式Runge-Kutta格式.数值例子显示,方法具有良好的激波分辨能力和高精度的数值逼近能力.     

Geometric gradient-flow dynamics with singular solutions

The gradient-flow dynamics of an arbitrary geometric quantity is derived using a generalization of Darcy’s Law. We consider flows in both Lagrangian and Eulerian formulations. The Lagrangian formulation includes a dissipative modification of fluid mechanics. Eulerian equations for self-organization of scalars, 1-forms and 2-forms are shown to reduce to nonlocal characteristic equations. We identify singular solutions of these equations corresponding to collapsed (clumped) states and discuss their evolution.     

An all-scale anelastic model for geophysical flows: dynamic grid deformation

We have developed an adaptive grid-refinement approach for simulating geophysical flows on scales from micro to planetary. Our model is nonoscillatory forward-in-time (NFT), nonhydrostatic, and anelastic. The major focus in this effort to date has been the design of a generalized mathematical framework for the implementation of deformable coordinates and its efficient numerical coding in a generic Eulerian/semi-Lagrangian NFT format. The key prerequisite of the adaptive grid is a time-dependent coordinate transformation, implemented rigorously throughout the governing equations of the model. The transformation enables mesh refinement indirectly via dynamic change of the metric coefficients, while retaining advantages of Cartesian mesh calculations (speed, low memory requirements, and accuracy) conducted fully in the computational domain. Diverse test results presented in this paper – simulations of a traveling stratospheric inertio-gravity-wave packet (with numerically advected dense-mesh region) and an idealized climate of the Earth (with analytically prescribed adaptive transformations) – demonstrate the potential and the efficacy of the new deformable grid model for tracing targeted flow features and dynamically adjusting to prescribed undulations of model boundaries.     

Lagrangian averages,averaged Lagrangians,and the mean effects of fluctuations in fluid dynamics

We begin by placing the generalized Lagrangian mean (GLM) equations for a compressible adiabatic fluid into the Euler-Poincare (EP) variational framework of fluid dynamics, for an averaged Lagrangian. This is the Lagrangian averaged Euler-Poincare (LAEP) theorem. Next, we derive a set of approximate small amplitude GLM equations (glm equations) at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the glm equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The glm derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction. Next, the new glm EP motion equations for incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM (or glm) fluid theory with a Taylor hypothesis closure. Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha equations. Thus, by using the LAEP theorem, we bridge between the GLM equations and the Euler-alpha closure equations, through the small-amplitude glm approximation in the EP variational framework. We conclude by highlighting a new application of the GLM, glm, and alpha-model results for Lagrangian averaged ideal magnetohydrodynamics. (c) 2002 American Institute of Physics.     

A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction

We present a new cell-centered multi-material arbitrary Lagrangian–Eulerian (ALE) scheme to solve the compressible gas dynamics equations on two-dimensional unstructured grid. Our ALE method is of the explicit time-marching Lagrange plus remap type. Namely, it involves the following three phases: a Lagrangian phase wherein the flow is advanced using a cell-centered scheme; a rezone phase in which the nodes of the computational grid are moved to more optimal positions; a cell-centered remap phase which consists of interpolating conservatively the Lagrangian solution onto the rezoned grid. The multi-material modeling utilizes either concentration equations for miscible fluids or the Volume Of Fluid (VOF) capability with interface reconstruction for immiscible fluids. The main original feature of this ALE scheme lies in the introduction of a new mesh relaxation procedure which keeps the rezoned grid as close as possible to the Lagrangian one. In this formalism, the rezoned grid is defined as a convex combination between the Lagrangian grid and the grid resulting from condition number smoothing. This convex combination is constructed through the use of a scalar parameter which is a scalar function of the invariants of the Cauchy–Green tensor over the Lagrangian phase. Regarding the cell-centered remap phase, we employ two classical methods based on a partition of the rezoned cell in terms of its overlap with the Lagrangian cells. The first one is a simplified swept face-based method whereas the second one is a cell-intersection-based method. Our multi-material ALE methodology is assessed through several demanding two-dimensional tests. The corresponding numerical results provide a clear evidence of the robustness and the accuracy of this new scheme.     

Remapping-free ALE-type kinetic method for flow computations

Based on the integral form of the fluid dynamic equations, a finite volume kinetic scheme with arbitrary control volume and mesh velocity is developed. Different from the earlier unified moving mesh gas-kinetic method [C.Q. Jin, K. Xu, An unified moving grid gas-kinetic method in Eulerian space for viscous flow computation, J. Comput. Phys. 222 (2007) 155–175], the coupling of the fluid equations and geometrical conservation laws has been removed in order to make the scheme applicable for any quadrilateral or unstructured mesh rather than parallelogram in 2D case. Since a purely Lagrangian method is always associated with mesh entangling, in order to avoid computational collapsing in multidimensional flow simulation, the mesh velocity is constructed by considering both fluid velocity (Lagrangian methodology) and diffusive velocity (Regenerating Eulerian mesh function). Therefore, we obtain a generalized Arbitrary-Lagrangian–Eulerian (ALE) method by properly designing a mesh velocity instead of re-generating a new mesh after distortion. As a result, the remapping step to interpolate flow variables from old mesh to new mesh is avoided. The current method provides a general framework, which can be considered as a remapping-free ALE-type method. Since there is great freedom in choosing mesh velocity, in order to improve the accuracy and robustness of the method, the adaptive moving mesh method [H.Z. Tang, T. Tang, Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41 (2003) 487–515] can be also used to construct a mesh velocity to concentrate mesh to regions with high flow gradients.