Semi-discrete Problem Research Articles

An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of the boundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming (SQP) algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.

The singular dynamic method for dynamic contact of thin elastic structures

This paper adresses the approximation of the dynamic impact of thin elastic structures. The principle of the presented method is the use of a singular mass matrix obtained by different discretizations of the deflection and velocity. The obtained semi-discretized problem is proved to be well-posed and energy conserving. The method is applied on some membrane, beam and plate models and associated numerical experiments are discussed.

An approximation scheme for a Hamilton–Jacobi equation defined on a network

In this paper we study approximation of Hamilton–Jacobi equations defined on a network. We introduce an appropriate notion of viscosity solution on networks which satisfies existence, uniqueness and stability properties. Then we define an approximation scheme of semi-Lagrangian type by discretizing in time the representation formula for the solution of Hamilton–Jacobi equations and we prove that the discrete problem admits a unique solution. Moreover we prove that the solution of the approximation scheme converges to the solution of the continuous problem uniformly on the network.In the second part of the paper we study a fully discrete scheme obtained via a finite elements discretization of the semi-discrete problem. Also for fully discrete scheme we prove the well posedness and the convergence to the viscosity solution of the Hamilton–Jacobi equation. We also discuss some issues concerning the implementation of the algorithm and we present some numerical examples.

Time–space adaptive numerical methods for the simulation of combustion fronts

This paper presents a new computational strategy for the simulation of combustion fronts based on adaptive time operator splitting and spatial multiresolution. High-order and dedicated one-step solvers compose the splitting scheme for the reaction, diffusion, and convection subproblems, to independently cope with their inherent numerical difficulties and to properly solve the corresponding temporal scales. Adaptive and thus highly compressed spatial representations for localized fronts originating from multiresolution analysis result in important reductions of memory usage, and hence numerical simulations with sufficiently fine spatial resolution can be performed with standard computational resources. The computational efficiency is further enhanced by splitting time steps established beyond standard stability constraints associated to mesh size or stiff source time scales. The splitting time steps are chosen according to a dynamic splitting technique relying on solid mathematical foundations, which ensures error control of the time integration and successfully discriminates time-varying multi-scale physics. For a given semi-discretized problem, the solution scheme provides dynamic accuracy estimates that reflect the quality of numerical results in terms of numerical errors of integration and compressed spatial representations, for general multi-dimensional problems modeled by stiff PDEs. The strategy is efficiently applied to simulate the propagation of laminar premixed flames interacting with vortex structures, as well as various configurations of self-ignition processes of diffusion flames in similar vortical hydrodynamics fields. A detailed study of the error control is provided and show the potential of the approach. It yields large gains in CPU time, while consistently describing a broad spectrum of space and time scales as well as different physical scenarios.

A Robust Spectral Method for Solving Heston’s Model

In this paper, we consider the Heston’s volatility model (Heston in Rev. Financ. Stud. 6: 327–343, 1993]. We simulate this model using a combination of the spectral collocation method and the Laplace transforms method. To approximate the two dimensional PDE, we construct a grid which is the tensor product of the two grids, each of which is based on the Chebyshev points in the two spacial directions. The resulting semi-discrete problem is then solved by applying the Laplace transform method based on Talbot’s idea of deformation of the contour integral (Talbot in IMA J. Appl. Math. 23(1): 97–120, 1979).

Error estimates for Galerkin approximations of the “classical” Boussinesq system

We consider the ‘classical’ Boussinesq system in one space dimension and its symmetric analog. These systems model two-way propagation of nonlinear, dispersive long waves of small amplitude on the surface of an ideal fluid in a uniform horizontal channel. We discretize an initial-boundary-value problem for these systems in space using Galerkin-finite element methods and prove error estimates for the resulting semidiscrete problems and also for their fully discrete analogs effected by explicit Runge-Kutta time-stepping procedures. The theoretical orders of convergence obtained are consistent with the results of numerical experiments that are also presented.

The balancing principle in solving semi-discrete inverse problems in Sobolev scales by Tikhonov method

In this article we discuss a regularization of semi-discrete ill-posed problem appearing as a result of application of a collocation method to Fredholm integral equation of the first kind. In this context we analyse Tikhonov regularization in Sobolev scales and prove error bounds under general source conditions. Moreover, we study an a posteriori regularization parameter choice by means of the balancing principle.

A PROJECTION-BASED VARIATIONAL MULTISCALE METHOD FOR LARGE-EDDY SIMULATION WITH APPLICATION TO NON-ISOTHERMAL FREE CONVECTION PROBLEMS

We consider a projection-based variational multiscale method for large-eddy simulation of the Navier–Stokes/Fourier model of incompressible, non-isothermal flows. For the semidiscrete problem, an a priori error estimate is given for rather general nonlinear, piecewise constant coefficients of the subgrid models for the unresolved scales of velocity, pressure, and temperature. Then we address aspects of the discretization in time. Finally, the design of the subgrid scale models is specified for the case of free convection problems and studied for the standard benchmark problem of free convection in a closed cavity.

A high order HODIE finite difference scheme for 1D parabolic singularly perturbed reaction–diffusion problems

This paper deals with the numerical approximation of the solution of 1D parabolic singularly perturbed problems of reaction–diffusion type. The numerical method combines the standard implicit Euler method on a uniform mesh to discretize in time and a HODIE compact fourth order finite difference scheme to discretize in space, which is defined on a priori special meshes condensing the grid points in the boundary layer regions. The method is uniformly convergent having first order in time and almost fourth order in space. The analysis of the uniform convergence is made in two steps, splitting the contribution to the error from the time and the space discretization. Although this idea has been previously used to prove the uniform convergence for parabolic singularly perturbed problems, here the proof is based on a new study of the asymptotic behavior of the exact solution of the semidiscrete problems obtained after the time discretization by using the Euler method. Some numerical results are given corroborating in practice the theoretical results.

Mixed finite element methods for a fourth order reaction diffusion equation

AbstractMixed finite element methods are applied to a fourth order reaction diffusion equation with different types of boundary conditions. Some a priori bounds are established with the help of Lyapunov functional. The semidiscrete schemes are derived using C0‐piecewise linear finite elements in spatial direction and error estimates are obtained. The semidiscrete problem is then discretized in the temporal direction using backward Euler method and the wellposedness of the completely discrete scheme is discussed. Finally, a priori error estimates are established. While deriving a priori error estimates, Gronwall's lemma is applied and the constants involved in the error bounds do not depend exponentially on \documentclass{article} \usepackage{amsmath, amsthm, amssymb, amsfonts}\pagestyle{empty}\begin{document}$\frac{1}{\gamma}$ \end{document}, where γ is a parameter appeared in the fourth order derivative. © 2011Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012

A Well-Posed Semi-Discretization of Elastodynamic Contact Problems with Friction

It has been shown numerically in previous works that the well-posedness of the spatial semi-discretization plays a crucial role in obtaining stable numerical schemes for elastodynamic frictionless contact problems. The purpose of this paper is thus to introduce a mass redistribution method adapted to elastodynamic contact with Coulomb friction that guarantees the well-posedness of the semi-discrete problem. It is shown that a differentiated treatment has to be applied to the friction condition. Some numerical tests illustrating the gain in stability for the midpoint time integration scheme are presented. They suggest also that, although the differentiated treatment is necessary for the well-posedness, it is not always mandatory from the numerical viewpoint.

Mathematical analysis to a nonlinear fourth-order partial differential equation

The paper first study the steady-state thin film type equation ∇ ⋅ ( u n | ∇ Δ u | q − 2 ∇ Δ u ) − δ u m Δ u = f ( x , u ) with Navier boundary conditions in multidimensional space. By the truncation method, a fixed point argument and some energy estimates, the existence and asymptotic limit δ → 0 for the positive weak solutions are given. Second, the parabolic equation u t + ( u n | u x x x | q − 2 u x x x ) x − δ u m u x x = 0 with a Navier boundary in one-dimensional space is researched. The existence is obtained by applying a semi-discrete method for the time variable and solving the corresponding elliptic problem. The uniqueness is shown for q = 2 depending on an energy estimate. In addition, the iteration relation of the semi-discrete problem gives an exponential decay result for the time t → ∞ . The thin film equation, which is usually used to describe the motion of a very thin layer of viscous in compressible fluids along an inclined plane, is a class of nonlinear fourth-order parabolic equations and the maximum principle does not hold directly. For applying the classic theory of partial differential equation, the paper transforms the fourth-order problem into a second-order elliptic–elliptic system or a second-order parabolic–elliptic system.

Discontinuous Galerkin time stepping with local projection stabilization for transient convection–diffusion-reaction problems

A time-dependent convection–diffusion-reaction problem is discretized in space by a continuous finite element method with local projection stabilization and in time by a discontinuous Galerkin method. We present error estimates for the semidiscrete problem after discretizing in space only and for the fully discrete problem. Numerical tests confirm the theoretical results.

Analysis of the Modified Mass Method for the Dynamic Signorini Problem with Coulomb Friction

The aim of the present work is to analyze the modified mass method for the dynamic Signorini problem with Coulomb friction. We prove that the space semidiscrete problem is equivalent to an upper semicontinuous one-sided Lipschitz differential inclusion and is, therefore, well-posed. We derive an energy balance. Next, considering an implicit time-integration scheme, we prove that, under a CFL-type condition on the discretization parameters, the fully discrete problem is well-posed. For a fixed discretization in space, we also prove that the fully discrete solutions converge to the space semidiscrete solution when the time step tends to zero.

Existence and asymptotic behavior of solutions to a thin film equation with Dirichlet boundary

This paper is devoted to studying a fourth-order parabolic equation u t + ε ( u n u x x x ) x − δ u x x = 0 with Dirichlet boundary. By transforming the fourth order problem into an elliptic-parabolic system and applying a semidiscretization method, the existence, uniqueness and regularization of solutions are obtained. By introducing an energy functional through the semi-discrete problem, we use the iteration method to show that the solutions of the developing equation exponentially converge to a constant steady state solution as the time variable t → ∞ . Finally, we get the result of the asymptotic limit δ → 0 .

Stable numerical schemes for a partly convolutional partial integro-differential equation

A model partial integro-differential operator (PIDO) that contains both local and nonlocal diffusion operators is considered in this article. This type of operators come in modeling various scientific and financial engineering problems. In most cases, people use finite difference schemes to generate solutions of such model problems. We compare and analyze stability and accuracy of two such finite difference schemes. We first present a discrete analogue of the PIDO and then approximate the semi-discrete time dependent problem using two different one step methods and show the stability conditions and the accuracy of the schemes. We use the Fourier transforms throughout our analysis.

Domain decomposition operator splitting for mimetic finite difference discretizations of non-stationary problems

In this paper, we propose reliable and efficient numerical methods for solving semilinear, time-dependent partial differential equations of reaction–diffusion type. The original problem is first integrated in time by using a linearly implicit fractional step Runge–Kutta method. This method takes advantage of a suitable partitioning of the diffusion operator based on domain decomposition techniques. The resulting semidiscrete problem is fully discretized by means of a mimetic finite difference method on quadrilateral meshes. Due to the previous splitting, the totally discrete scheme can be reduced to a set of uncoupled linear systems which can be solved in parallel. The overall algorithm is unconditionally stable and second-order convergent in both time and space. These properties are confirmed by numerical experiments.

Error analysis of thermal equation with moving ends

A mathematical model of the linear thermodynamic equations with moving ends, based on the Stefan Problem is considered. In this work, we are interested in obtaining existence, uniqueness and regularity using the Galerkin method and mainly to establish an error estimates of solutions in Sobolev spaces for the semi-discrete problem, with discretization of space variable, continuous time and for the completely discrete problem with discretization of space variable and time.

A stable and high-order accurate conjugate heat transfer problem

This paper analyzes well-posedness and stability of a conjugate heat transfer problem in one space dimension. We study a model problem for heat transfer between a fluid and a solid. The energy method is used to derive boundary and interface conditions that make the continuous problem well-posed and the semi-discrete problem stable. The numerical scheme is implemented using 2nd-, 3rd- and 4th-order finite difference operators on Summation-By-Parts (SBP) form. The boundary and interface conditions are implemented weakly. We investigate the spectrum of the spatial discretization to determine which type of coupling that gives attractive convergence properties. The rate of convergence is verified using the method of manufactured solutions.

Mixed finite element method for electrowetting on dielectric with contact line pinning

We present a mixed finite element method for a model of the flow in a Hele-Shaw cell of 2-D fluid droplets surrounded by air driven by surface tension and actuated by an electric field. The application of interest regards a micro-fluidic device called ElectroWetting on Dielectric (EWOD). Our analysis first focuses on the time discrete (continuous in space) problem and is presented in a mixed variational framework, which incorporates curvature as a natural boundary condition. The model includes a viscous damping term for interface motion, as well as contact line pinning (sticking of the interface) and is captured in our formulation by a variational inequality. The semi-discrete problem uses a semiimplicit time discretization of curvature. We prove the well-posedness of the semi-discrete problem and fully discrete problem when discretized with iso-parametric finite elements. We derive a priori error estimates for the space discretization. We also prove the convergence of an Uzawa algorithm for solving the semi-discrete EWOD system with inequality constraint. We conclude with a discussion about experimental orders of convergence.