Simulation of flows with strong shocks with an adaptive conservative scheme

In the present work, numerical simulations of unsteady flows with moving shocks are presented. An unsteady mesh adaptation method, based on error equidistribution criteria, is adopted to capture the most important flow features. The modifications to the topology of the grid are locally interpreted in terms of continuous deformation of the finite volumes built around the nodes. The arbitrary Lagrangian–Eulerian formulation of the Euler equations is then applied to compute the flow variable over the new grid without resorting to any explicit interpolation step. The numerical results show an increase in the accuracy of the solution, together with a strong reduction of the computational costs, with respect to computations with a uniform grid using a larger number of nodes.     

A Comparative Numerical Study of Meshing Functionals for Variational Mesh Adaptation

We present a comparative numerical study for three functionals used for variational mesh adaptation. One of them is a generalization of Winslow's variable diffusion functional while the others are based on equidistribution and alignment. These functionals are known to have nice theoretical properties and work well for most mesh adaptation problems either as a stand-alone variational method or combined within the moving mesh framework. Their performance is investigated numerically in terms of equidistribution and alignment mesh quality measures. Numerical results in 2D and 3D are presented.     

Optimal adaptive grids of least-squares finite element methods in two spatial dimensions

This article concerns a procedure to generate optimal adaptive grids for convection dominated problems in two spatial dimensions based on least-squares finite element approximations. The procedure extends a one dimensional equidistribution principle which minimizes the interpolation error in some norms. The idea is to select two directions which can reflect the physics of the problems and then apply the one dimensional equidistribution principle to the chosen directions. Model problems considered are the two dimensional convection-diffusion problems where boundary and interior layers occur. Numerical results of model problems illustrating the efficiency of the proposed scheme are presented. In addition, to avoid skewed mesh in the optimal grids generated by the algorithm, an unstructured local mesh smoothing will be considered in the least-squares approximations. Comparisons with the Gakerkin finite element method will also be provided.     

An ultra-weak method for acoustic fluid–solid interaction

We introduce the ultra-weak variational formulation (UWVF) for fluid–solid vibration problems. In particular, we consider the scattering of time-harmonic acoustic pressure waves from solid, elastic objects. The problem is modeled using a coupled system of the Helmholtz and Navier equations. The transmission conditions on the fluid–solid interface are represented in an impedance-type form after which we can employ the well known ultra-weak formulations for the Helmholtz and Navier equations. The UWVF approximation for both equations is computed using a superposition of propagating plane waves. A condition number based criterion is used to define the plane wave basis dimension for each element. As a model problem we investigate the scattering of sound from an infinite elastic cylinder immersed in a fluid. A comparison of the UWVF approximation with the analytical solution shows that the method provides a means for solving wave problems on relatively coarse meshes. However, particular care is needed when the method is used for problems at frequencies near the resonance frequencies of the fluid–solid system.     

A Simple and Effective Self-adaptive Moving Mesh for Enthalpy Formulations of Phase Change Problems

A self-adaptive moving mesh for enthalpy formulations of diffusiondriven phase change problems is described. The PDE is approximatedby a finite element method on a moving, irregular mesh. Themesh is determined by a novel equidistribution principle andis concentrated in the phase transition regions and uniformin regions of pure phase.     

The least squares spectral element method for the Cahn–Hilliard equation

The problem of numerically resolving an interface separating two different components is a common problem in several scientific and engineering applications. One alternative is to use phase field or diffuse interface methods such as the Cahn–Hilliard (C–H) equation, which introduce a continuous transition region between the two bulk phases. Different numerical schemes to solve the C–H equation have been suggested in the literature. In this work, the least squares spectral element method (LS-SEM) is used to solve the Cahn–Hilliard equation. The LS-SEM is combined with a time–space coupled formulation and a high order continuity approximation by employing C¹¹p-version hierarchical interpolation functions both in space and time. A one-dimensional case of the Cahn–Hilliard equation is solved and the convergence properties of the presented method analyzed. The obtained solution is in accordance with previous results from the literature and the basic properties of the C–H equation (i.e. mass conservation and energy dissipation) are maintained. By using the LS-SEM, a symmetric positive definite problem is always obtained, making it possible to use highly efficient solvers for this kind of problems. The use of dynamic adjustment of number of elements and order of approximation gives the possibility of a dynamic meshing procedure for a better resolution in the areas close to interfaces.     

Adaptive finite element methods for Cahn–Hilliard equations

We develop a method for adaptive mesh refinement for steady state problems that arise in the numerical solution of Cahn–Hilliard equations with an obstacle free energy. The problem is discretized in time by the backward-Euler method and in space by linear finite elements. The adaptive mesh refinement is performed using residual based a posteriori estimates; the time step is adapted using a heuristic criterion. We describe the space–time adaptive algorithm and present numerical experiments in two and three space dimensions that demonstrate the usefulness of our approach.     

Numerical simulation of blowup in nonlocal reaction–diffusion equations using a moving mesh method

In this paper we implement the moving mesh PDE method for simulating the blowup in reaction–diffusion equations with temporal and spacial nonlinear nonlocal terms. By a time-dependent transformation, the physical equation is written into a Lagrangian form with respect to the computational variables. The time-dependent transformation function satisfies a parabolic partial differential equation — usually called moving mesh PDE (MMPDE). The transformed physical equation and MMPDE are solved alternately by central finite difference method combined with a backward time-stepping scheme. The integration time steps are chosen to be adaptive to the blowup solution by employing a simple and efficient approach. The monitor function in MMPDEs plays a key role in the performance of the moving mesh PDE method. The dominance of equidistribution is utilized to select the monitor functions and a formal analysis is performed to check the principle. A variety of numerical examples show that the blowup profiles can be expressed correctly in the computational coordinates and the blowup rates are determined by the tests.     

Anisotropic mesh adaptation for numerical solution of boundary value problems

We present an efficient mesh adaptation algorithm that can be successfully applied to numerical solutions of a wide range of 2D problems of physics and engineering described by partial differential equations. We are interested in the numerical solution of a general boundary value problem discretized on triangular grids. We formulate a necessary condition for properties of the triangulation on which the discretization error is below the prescribed tolerance and control this necessary condition by the interpolation error. For a sufficiently smooth function, we recall the strategy how to construct the mesh on which the interpolation error is below the prescribed tolerance. Solving the boundary value problem we apply this strategy to the smoothed approximate solution. The novelty of the method lies in the smoothing procedure that, followed by the anisotropic mesh adaptation (AMA) algorithm, leads to the significant improvement of numerical results. We apply AMA to the numerical solution of an elliptic equation where the exact solution is known and demonstrate practical aspects of the adaptation procedure: how to control the ratio between the longest and the shortest edge of the triangulation and how to control the transition of the coarsest part of the mesh to the finest one if the two length scales of all the triangles are clearly different. An example of the use of AMA for the physically relevant numerical simulation of a geometrically challenging industrial problem (inviscid transonic flow around NACA0012 profile) is presented. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

A $\theta$-$L$ Approach for Solving Solid-State Dewetting Problems

We propose a $\theta$-$L$ approach for solving a sharp-interface model about simulating solid-state dewetting of thin films with isotropic/weakly anisotropic surface energies. The sharp-interface model is governed by surface diffusion and contact line migration. For solving the model, traditional numerical methods usually suffer from the severe stability constraint and/or the mesh distribution trouble. In the $\theta$-$L$ approach, we introduce a useful tangential velocity along the evolving interface and utilize a new set of variables (i.e., the tangential angle $\theta$ and the total length $L$ of the interface curve), so that it not only could reduce the stiffness resulted from the surface tension, but also could ensure the mesh equidistribution property during the evolution. Furthermore, it can achieve second-order accuracy when implemented by a semi-implicit linear finite element method. Numerical results are reported to demonstrate that the proposed $\theta$-$L$ approach is efficient and accurate.     

On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid

We consider the motion of a rigid body immersed in a bidimensional incompressible perfect fluid. The motion of the fluid is governed by the Euler equations and the conservation laws of linear and angular momentum rule the dynamics of the rigid body. We prove the existence and uniqueness of a global classical solution for this fluid–structure interaction problem. The proof relies mainly on weighted estimates for the vorticity associated with the strong solution of a fluid–structure interaction problem obtained by incorporating some viscosity.     

Optimal error estimate of upwind scheme on Shishkin-type meshes for singularly perturbed parabolic problems with discontinuous convection coefficients

This paper analyzes the implicit upwind finite difference scheme on Shishkin-type meshes (including the classical piecewise-uniform Shishkin mesh and the Bakhalov-Shishkin mesh) for a class of singularly perturbed parabolic convection-diffusion problems exhibiting strong interior layers. Suitable conditions on the mesh-generating functions are derived and are found to be sufficient for the convergence of the method, uniformly with respect to the perturbation parameter. Utilizing these conditions, it is shown that the method converges uniformly in the discrete supremum norm with an optimal error bound. Numerical results are presented to validate the theoretical results.     

Upwind Based Parameter Uniform Convergence Analysis for Two Parametric Parabolic Convection Diffusion Problems by Moving Mesh Methods

A parabolic two parametric convection-diffusion reaction problem is considered for the moving mesh error analysis. The continuous problem is discretized by the first order upwind scheme on a non uniform mesh. A curvature based error monitor function is proposed to generate the layer adapted mesh. It is proved that the numerical solution converges to the exact solution on the mesh obtained by the equidistribution of the proposed monitor function. The convergence is first order accurate. The present analysis generalizes the results obtained in earlier publications [8,9]. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A spatial domain decomposition method for parabolic optimal control problems

We present a non-overlapping spatial domain decomposition method for the solution of linear–quadratic parabolic optimal control problems. The spatial domain is decomposed into non-overlapping subdomains. The original parabolic optimal control problem is decomposed into smaller problems posed on space–time cylinder subdomains with auxiliary state and adjoint variables imposed as Dirichlet boundary conditions on the space–time interface boundary. The subdomain problems are coupled through Robin transmission conditions. This leads to a Schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space-time interface boundary. The Schur complement operator is the sum of space–time subdomain Schur complement operators. The application of these subdomain Schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem. The subdomain Schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem. We introduce a new family of Neumann–Neumann type preconditioners for the Schur complement system including several different coarse grid corrections. We compare the numerical performance of our preconditioners with an alternative approach recently introduced by Benamou.     

扩展型动网格的Chebyshev有限谱方法

给出了基于非均匀网格的Chebyshev有限谱方法．提出了可生成两种类型扩展型动网格的均布格式．一种类型的网格被用来提高波面附近的分辨率,另一种类型则用在梯度较大的流动区域．由于采用Chebyshev多项式作为基函数,该方法具有高阶精度．从上个时间步到当前时间步,两套不均匀网格间的物理量采用Chebyshev多项式插值．为使方法在时间离散方面保持高精度,采用了Adams-Bashforth预报格式和Adams-Moulton校正格式．为了避免由Korteweg-deVries(KdV)方程的弥散项引起的数值振荡,给出了一种非均匀网格下的数值稳定器．给出的方法与具有分析解的Burgers方程的非线性对流扩散问题和KdV方程的单孤独波和双孤独波传播问题进行了比较,结果非常吻合．     

Maximum principle for the finite element solution of time‐dependent anisotropic diffusion problems

Preservation of the maximum principle is studied for the combination of the linear finite element method in space and the θ ‐method in time for solving time‐dependent anisotropic diffusion problems. It is shown that the numerical solution satisfies a discrete maximum principle when all element angles of the mesh measured in the metric specified by the inverse of the diffusion matrix are nonobtuse, and the time step size is bounded below and above by bounds proportional essentially to the square of the maximal element diameter. The lower bound requirement can be removed when a lumped mass matrix is used. In two dimensions, the mesh and time step conditions can be replaced by weaker Delaunay‐type conditions. Numerical results are presented to verify the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A second-order hybrid finite difference scheme for a system of singularly perturbed initial value problems

A system of coupled singularly perturbed initial value problems with two small parameters is considered. The leading term of each equation is multiplied by a small positive parameter, but these parameters may have different magnitudes. The solution of the system has boundary layers that overlap and interact. The structure of these layers is analyzed, and this leads to the construction of a piecewise-uniform mesh that is a variant of the usual Shishkin mesh. On this mesh a hybrid finite difference scheme is proved to be almost second-order accurate, uniformly in both small parameters. Numerical results supporting the theory are presented.     

Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers

A numerical method is proposed for solving singularly perturbed turning point problems exhibiting twin boundary layers based on the reproducing kernel method (RKM). The original problem is reduced to two boundary layers problems and a regular domain problem. The regular domain problem is solved by using the RKM. Two boundary layers problems are treated by combining the method of stretching variable and the RKM. The boundary conditions at transition points are obtained by using the continuity of the approximate solution and its first derivatives at these points. Two numerical examples are provided to illustrate the effectiveness of the present method. The results compared with other methods show that the present method can provide very accurate approximate solutions.     

A higher order difference method for singularly perturbed parabolic partial differential equations

This paper studies a higher order numerical method for the singularly perturbed parabolic convection-diffusion problems where the diffusion term is multiplied by a small perturbation parameter. In general, the solutions of these type of problems have a boundary layer. Here, we generate a spatial adaptive mesh based on the equidistribution of a positive monitor function. Implicit Euler method is used to discretize the time variable and an upwind scheme is considered in space direction. A higher order convergent solution with respect to space and time is obtained using the postprocessing based extrapolation approach. It is observed that the convergence is independent of perturbation parameter. This technique enhances the order of accuracy from first order uniform convergence to second order uniform convergence in space as well as in time. Comparative study with the existed meshes show the highly effective behavior of the present method.