Level Set Calculations for Incompressible Two-Phase Flows on a Dynamically Adaptive Grid

We present a coupled moving mesh and level set method for computing incompressible two-phase flow with surface tension. This work extends a recent work of Di et al. [(2005). SIAM J. Sci. Comput. 26, 1036–1056] where a moving mesh strategy was proposed to solve the incompressible Navier–Stokes equations. With the involvement of the level set function and the curvature of the interface, some subtle issues in the moving mesh scheme, in particular the solution interpolation from the old mesh to the new mesh and the choice of monitor functions, require careful considerations. In this work, a simple monitor function is proposed that involves both the level set function and its curvature. The purpose for designing the coupled moving mesh and level set method is to achieve higher resolution for the free surface by using a minimum amount of additional expense. Numerical experiments for air bubbles and water drops are presented to demonstrate the effectiveness of the proposed scheme.     

H-adaptive Mesh Method with Double Tolerance Adaptive Strategy for Hyperbolic Conservation Laws

The numerical method used to solve hyperbolic conservation laws is often an explicit scheme. As a commonly used technique to improve the quality of numerical simulation, the $h$ -adaptive mesh method is adopted to resolve sharp structures in the solution. Since the computational costs of altering the mesh and solving the PDEs are comparable, too often the mesh adaption triggered may bring down the overall efficiency of solving hyperbolic conservation laws using $h$ -adaptive mesh method. In this paper, we propose a so-called double tolerance adaptive strategy to optimize the overall numerical efficiency by reducing the number of mesh adaptions, as well as preserving the quality of the numerical solution. Numerical results are presented to demonstrate the robustness and effectiveness of our $h$ -adaptive algorithm using the double tolerance adaptive strategy.     

Strong Stability Preserving Explicit Peer Methods for Discontinuous Galerkin Discretizations

In this paper we study explicit peer methods with the strong stability preserving (SSP) property for the numerical solution of hyperbolic conservation laws in one space dimension. A system of ordinary differential equations is obtained by discontinuous Galerkin (DG) spatial discretizations, which are often used in the method of lines approach to solve hyperbolic differential equations. We present in this work the construction of explicit peer methods with stability regions that are designed for DG spatial discretizations and with large SSP coefficients. Methods of second- and third order with up to six stages are optimized with respect to both properties. The methods constructed are tested and compared with appropriate Runge–Kutta methods. The advantage of high stage order is verified numerically.     

变形法移动网格求解双曲型守恒律方程研究

在现有格式的基础上要提高偏微分方程数值解的分辨率,自适应移动网格技术是一种有效而且可行的方法。文中将文献[1]提出的自适应移动网格技术推广到三角形网格,并将该方法用于求解双曲型守恒量方程。用网格自适应技术求解守恒律问题时,当生成新网格之后,需要将旧网格上的函数值更新到新的网格,并保持物理量的守恒性。针对这个问题,文中提出了函数值更新过程中守恒型插值公式的具体形式,并针对二维双曲型守恒律方程进行了仿真实验,取得了满意的结果。     

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

We explore the use of radial basis functions (RBF) in the weighted essentially non-oscillatory (WENO) reconstruction process used to solve hyperbolic conservation laws, resulting in a numerical method of arbitrarily high order to solve problems with discontinuous solutions. Thanks to the mesh-less property of the RBFs, the method is suitable for non-uniform grids and mesh adaptation. We focus on multiquadric radial basis functions and propose a simple strategy to choose the shape parameter to control the balance between achievable accuracy and the numerical stability. We also develop an original smoothness indicator which is independent of the RBF for the WENO reconstruction step. Moreover, we introduce type I and type II RBF-WENO methods by computing specific linear weights. The RBF-WENO method is used to solve linear and nonlinear problems for both scalar and systems of conservation laws, including Burgers equation, the Buckley–Leverett equation, and the Euler equations. Numerical results confirm the performance of the proposed method. We finally consider an effective conservative adaptive algorithm that captures moving shocks and rapidly varying solutions well. Numerical results on moving grids are presented for both Burgers equation and the more complex Euler equations.     

Second-Order Accurate Godunov Scheme for Multicomponent Flows on Moving Triangular Meshes

This paper presents a second-order accurate adaptive Godunov method for two-dimensional (2D) compressible multicomponent flows, which is an extension of the previous adaptive moving mesh method of Tang et al. (SIAM J. Numer. Anal. 41:487–515, 2003) to unstructured triangular meshes in place of the structured quadrangular meshes. The current algorithm solves the governing equations of 2D multicomponent flows and the finite-volume approximations of the mesh equations by a fully conservative, second-order accurate Godunov scheme and a relaxed Jacobi-type iteration, respectively. The geometry-based conservative interpolation is employed to remap the solutions from the old mesh to the newly resulting mesh, and a simple slope limiter and a new monitor function are chosen to obtain oscillation-free solutions, and track and resolve both small, local, and large solution gradients automatically. Several numerical experiments are conducted to demonstrate robustness and efficiency of the proposed method. They are a quasi-2D Riemann problem, the double-Mach reflection problem, the forward facing step problem, and two shock wave and bubble interaction problems.     

Three dimensional discontinuous Galerkin methods for Euler equations on adaptive conforming meshes

In the numerical simulation of three dimensional fluid dynamical equations, the huge computational quantity is a main challenge. In this paper, the discontinuous Galerkin (DG) finite element method combined with the adaptive mesh refinement (AMR) is studied to solve the three dimensional Euler equations based on conforming unstructured tetrahedron meshes, that is according the equation solution variation to refine and coarsen grids so as to decrease total mesh number. The four space adaptive strategies are given and analyzed their advantages and disadvantages. The numerical examples show the validity of our methods.     

The simulation of 3D unsteady incompressible flows with moving boundaries on unstructured meshes

The development of a computational model for the simulation of three-dimensional unsteady incompressible viscous fluid flows with moving boundaries is presented. The numerical model is based upon the solution of the Navier–Stokes equations on unstructured meshes using the artificial compressibility approach. An ALE formulation is adopted and the equations are discretized using a cell vertex finite volume method. The formulation ensures the satisfaction of the geometric conservation law when the mesh is allowed to move. An implicit time discretization is adopted and a dual time approach is employed. Explicit relaxation is used for the sub-iterations, with multigrid acceleration. For moving geometries, the mesh is deformed by adopting a spring analogy, combined with a wall distance function approach. The numerical procedure is validated on a standard problem and is then used for the simulation of flow over a flexible fish-like body.     

Moving Mesh Method with Error-Estimator-Based Monitor and Its Applications to Static Obstacle Problem

The main objective of this work is to demonstrate that sharp a posteriori error estimators can be employed as appropriate monitor functions for moving mesh methods. We illustrate the main ideas by considering elliptic obstacle problems. Some important issues such as how to derive the sharp estimators and how to smooth the monitor functions are addressed. The numerical schemes are applied to a number of test problems in two dimensions. It is shown that the moving mesh methods with the proposed monitor functions can effectively capture the free boundaries of the elliptic obstacle problems and reduce the numerical errors arising from the free boundaries.     

Moving mesh methods for solving parabolic partial differential equations

A new adaptive method is described for solving nonlinear parabolic partial differential equations with moving boundaries, using a moving mesh with continuous finite elements. The evolution of the mesh within the interior of the spatial domain is based upon conserving the distribution of a chosen monitor function across the domain throughout time, where the initial distribution is selected based upon the given initial data. The mesh movement at the boundary is governed by a second monitor function, which may or may not be the same as that used to drive the interior mesh movement. The method is described in detail and a selection of computational examples are presented using different monitor functions applied to the porous medium equation (PME) in one and two space dimensions.     

On Resistive MHD Models with Adaptive Moving Meshes

In this paper we describe an adaptive moving mesh technique and its application to convection-diffusion models from magnetohydrodynamics (MHD). The method is based on a coordinate transformation between physical and computational coordinates. The transformation can be viewed as a solution of adaptive mesh partial differential equations (PDEs) which are derived from the minimization of a mesh-energy integral. For an efficient implementation we have used an approach in which the numerical solution of the physical PDE model and the adaptive PDEs are decoupled. Further, to avoid solving large nonlinear systems, an implicit-explicit method is applied for the time integration in combination with the iterative method Bi-CGSTAB. The adaptive mesh can be viewed as a 2D variant of the equidistribution principle, and it has the ability to track individual features of the physical solutions in the developing plasma flows. The results of a series of numerical experiments are presented which cover several aspects typifying resistive magnetofluid-dynamics.     

Operator-splitting methods for the 2D convective Cahn–Hilliard equation

In this work, we present operator-splitting methods for the two-dimensional nonlinear fourth-order convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The full problem is split into hyperbolic, nonlinear diffusion and linear fourth-order problems. We prove that the semi-discrete approximate solution obtained from the operator-splitting method converges to the weak solution. Numerical methods are then constructed to solve each sub equations sequentially. The hyperbolic conservation law is solved by efficient finite volume methods and dimensional splitting method, while the one-dimensional hyperbolic conservation laws are solved using front tracking algorithm. The front tracking method is based on the exact solution and hence has no stability restriction on the size of the time step. The nonlinear diffusion problem is solved by a linearized implicit finite volume method, which is unconditionally stable. The linear fourth-order equation is solved using a pseudo-spectral method, which is based on an exact solution. Finally, some numerical experiments are carried out to test the performance of the proposed numerical methods.     

Comparative study of different numerical approaches in space–time CESE framework for high-fidelity flow simulations

With the advancement of computer hardware, the trend of research in computational fluid dynamics is moving towards development of highly accurate, unstructured-mesh compatible, robust and efficient numerical methods for simulating problems involving strong transient effects and relatively complex geometries as well as physics. The space–time conservation element and solution element method is a genuinely multi-dimensional, unstructured-mesh compatible numerical framework, which was built from a consistent and synergetic integration of conservation laws in the space–time domain to avoid the limitations of conventional schemes, such as the use of 1-D flux reconstruction with a Riemann solver. It has been shown that the framework can be used for time-accurate simulations of a variety of problems involving unsteady waves, strong flow discontinuities, and their interactions with remarkable accuracy. However, this method at its current state has encountered the challenge in balancing the robustness and numerical accuracy when highly stretched meshes were used in viscous flow simulation. In this paper, we briefly discuss various numerical approaches developed for this framework thus far as well as their strengths and weaknesses, and conduct a comparative study of their numerical accuracies using some 2-D viscous benchmark test cases. The application of this method in realistic, complex 3-D problems is also included here to demonstrate its computational efficiency in large-scale computing.     

A second-order TVD implicit–explicit finite volume method for time-dependent convection-reaction equations

A class of conservative methods is developed in the more general framework of cell-centered upwind differences to approximate numerically the solution of one-dimensional non-linear conservation laws with (possibly) stiff reaction source terms. These methods are based on a non-oscillatory piecewise linear polynomial representation of the discrete solution within any mesh interval to compute pointwise solution values. The piecewise linear approximate solution is obtained by approximating the cell average of the analytical solution and the solution slope in every mesh cell. These two quantities are evolved in time by solving a set of discrete equations that are suitably designed to ensure formal second-order consistency. Several numerical tests which are taken from literature illustrate the performance of the method in solving non-stiff and stiff convection-reaction equations in conservative form.     

Numerical Convergence Study of Nearly Incompressible,Inviscid Taylor–Green Vortex Flow

A spectral method and a fifth-order weighted essentially non-oscillatory method were used to examine the consequences of filtering in the numerical simulation of the three-dimensional evolution of nearly-incompressible, inviscid Taylor–Green vortex flow. It was found that numerical filtering using the high-order exponential filter and low-pass filter with sharp high mode cutoff applied in the spectral simulations significantly affects the convergence of the numerical solution. While the conservation property of the spectral method is highly desirable for fluid flows described by a system of hyperbolic conservation laws, spectral methods can yield erroneous results and conclusions at late evolution times when the flow eventually becomes under-resolved. In particular, it is demonstrated that the enstrophy and kinetic energy, which are two integral quantities often used to evaluate the quality of numerical schemes, can be misleading and should not be used unless one can assure that the solution is sufficiently well-resolved. In addition, it is shown that for the Taylor–Green vortex (for example) it is useful to compare the predictions of at least two numerical methods with different algorithmic foundations (such as a spectral and finite-difference method) in order to corroborate the conclusions from the numerical solutions when the analytical solution is not known.     

Adaptive moving mesh methods for two-dimensional resistive magneto-hydrodynamic PDE models

An adaptive moving mesh technique is applied to magneto-hydrodynamics (MHD) model problem. The moving mesh strategy is based on the approach proposed in Li et al. [Li R, Tang T, Zhang P. Moving mesh methods in multiple dimensions based on harmonic maps. J Comput Phys 2001;170:562-88] to separate the mesh-moving and PDE evolution at each time step. The Magneto-hydrodynamic equations are discretized by a finite-volume method in space, and the mesh-moving part is realized by solving the Euler-Lagrange equations to minimize a certain variation with the directional splitting monitor function. A conservative interpolation is used to redistribute the numerical solutions on the new meshes. Numerical results demonstrate the accuracy and effectiveness of the proposed algorithm.     

An Adaptive Moving Mesh Finite Element Solution of the Regularized Long Wave Equation

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a \(C^0\) finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution.     

A study of moving mesh methods applied to a thin flame propagating in a detonator delay element

We study the application of moving mesh methods to a one-dimensional (time dependent) detonator delay element problem. We consider moving mesh methods based on the equidistribution principle derived by Huang et al. [1]. Adaptive mesh methods have been widely used recently to solve time dependent partial differential equations having large solution gradients. Significant improvements in accuracy and efficiency are achieved by adapting the nodes (mesh points) so that they are concentrated about areas of large solution variations. Each system of equations for the moving mesh methods is solved in conjunction with the detonator problem. In this paper, the system of ordinary differential equations that results (after discretising in space) is solved using the double precision version of the stiff ordinary differential equation solver DASSL. The numerical results clearly demonstrate that the moving mesh methods are capable of tracking the deflagration wave as it travels down the detonator delay element more accurately and more efficiently than a fixed mesh method.     

Adaptive mesh refinement for chevron nozzle jet flows

In this paper an adaptive mesh generation procedure is presented for improving the resolution of the numerical simulation of a turbulent jet exhausting from a chevron nozzle. This procedure is based on the minimization of a variational integral whose integrand depends on the metric (also called the monitor function) induced by a curvilinear grid generated in the physical domain. Specifically, it leads to solving parabolic equations involving the monitor function, which is carefully designed to resolve the flow gradients, and which, in the present instance, is determined by the time-averaged axial velocity profile within the jet. This mesh redistribution strategy is incorporated into a flow computation code (that solves the compressible three-dimensional Navier-Stokes equations using a prefactored optimized fourth-order compact difference scheme for spatial derivatives and the Beam-Warming method for the time derivative on a multi-block overset grid) and is demonstrated to be efficient and effective.     

A posteriori error analysis for a fractional differential equation

Numerical treatment for a fractional differential equation (FDE) is proposed and analysed. The solution of the FDE may be singular near certain domain boundaries, which leads to numerical difficulty. We apply the upwind finite difference method to the FDE. The stability properties and a posteriori error analysis for the discrete scheme are given. Then, a posteriori adapted mesh based on a posteriori error analysis is established by equidistributing arc-length monitor function. Numerical experiments illustrate that the upwind finite difference method on a posteriori adapted mesh is more accurate than the method on uniform mesh.