A M\"untz-Collocation spectral method for weakly singular volterra integral equations

In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the second kind Volterra integral equations (VIEs) with weakly singular kernel $(x-s)^{-\mu},0<\mu<1$. First we develop a family of fractional Jacobi polynomials, along with basic approximation results for some weighted projection and interpolation operators defined in suitable weighted Sobolev spaces. Then we construct an efficient fractional Jacobi-collocation spectral method for the VIEs using the zeros of the new developed fractional Jacobi polynomial. A detailed convergence analysis is carried out to derive error estimates of the numerical solution in both $L^{\infty}$- and weighted $L^{2}$-norms. The main novelty of the paper is that the proposed method is highly efficient for typical solutions that VIEs usually possess. Precisely, it is proved that the exponential convergence rate can be achieved for solutions which are smooth after the variable change $x\rightarrow x^{1/\lambda}$ for a suitable real number $\lambda$. Finally a series of numerical examples are presented to demonstrate the efficiency of the method

An extension of the landweber regularization for a backward time fractional wave problem

In this paper, we investigate numerical methods for a backward problem of the time-fractional wave equation in bounded domains. We propose two fractional filter regularization methods, which can be regarded as an extension of the classical Landweber iteration for the time-fractional wave backward problem. The idea is first to transform the ill-posed backward problem into a weighted normal operator equation, then construct the regularization methods for the operator equation by introducing suitable fractional filters. Both a priori and a posteriori regularization parameter choice rules are investigated, together with an estimate for the smallest regularization parameter according to a discrepancy principle. Furthermore, an error analysis is carried out to derive the convergence rates of the regularized solutions generated by the proposed methods. The theoretical estimate shows that the proposed fractional regularizations efficiently overcome the well-known over-smoothing drawback caused by the classical regularizations. Some numerical examples are provided to confirm the theoretical results. In particular, our numerical tests demonstrate that the fractional regularization is actually more efficient than the classical methods for problems having low regularity.Méthode des champs : algorithmes et simulations de phénomènes complexesInitiative d'excellence de l'Université de Bordeau

New Unconditionally Stable Schemes for the Navier-Stokes Equations

In this paper we propose some efficient schemes for the Navier-Stokes equa- tions. The proposed schemes are constructed based on an auxiliary variable reformu- lation of the underlying equations, recently introduced by Li et al. [20]. Our objective is to construct and analyze improved schemes, which overcome some of the shortcom- ings of the existing schemes. In particular, our new schemes have the capability to capture steady solutions for large Reynolds numbers and time step sizes, while keeping the error analysis available. The novelty of our method is twofold: i) Use the Uzawa algorithm to decouple the pressure and the velocity. This is to replace the pressure- correction method considered in [20]. ii) Inspired by the paper [21], we modify the algorithm using an ingredient to capture stationary solutions. In all cases we ana- lyze a first- and second-order schemes and prove the unconditionally energy stability. We also provide an error analysis for the first-order scheme. Finally we validate our schemes by performing simulations of the Kovasznay flow and double lid driven cav- ity flow. These flow simulations at high Reynolds numbers demonstrate the robustness and efficiency of the proposed scheme

An efficient numerical approach for stochastic evolution PDEs driven by random diffusion coefficients and multiplicative noise

In this paper, we investigate the stochastic evolution equations (SEEs) driven by a bounded $\log$-Whittle-Mat$\acute{{\mathrm{e}}}$rn (W-M) random diffusion coefficient field and $Q$-Wiener multiplicative force noise. First, the well-posedness of the underlying equations is established by proving the existence, uniqueness, and stability of the mild solution. A sampling approach called approximation circulant embedding with padding is proposed to sample the random coefficient field. Then a spatio-temporal discretization method based on semi-implicit Euler-Maruyama scheme and finite element method is constructed and analyzed. An estimate for the strong convergence rate is derived. Numerical experiments are finally reported to confirm the theoretical result.</abstract

Semidiscrete approximation of the penalty approach to the stabilization of the Boussinesq system by localized feedback control

We study the numerical approximation of the stabilization of the semidiscrete linearized Boussinesq system around an unstable stationary state. The stabilization is achieved by internal feedback controls applied on the velocity and the temperature equations, localized in an arbitrary open subset. This article follows the framework of [11], considering the continuous linearized Boussinesq system. The goal is to study the approximation by penalization of the free divergence condition in the semidiscrete case. More precisely, considering infinite time horizon LQR optimal control problem, we establish convergence results for the optimal controls, optimal solutions and Riccati operators when the penalization parameter goes to zero. We then propose a numerical validation of these results in a two-dimensional setting

Convergence de la POD pour des problèmes multiparamètre

On s'intersse à l'analyse numérique de la convergence de la POD pour représenter  des solutions de l'équation de la chaleur paramétisée. La paramètre étant la condictivité.  A l'aide de plusiers exemples numériques nous montrons la convergence exponentielle en fonction  des modes POD. Ensuite nous apportons quelques éléments pour la justification mathématique de ce résultat

Computers & Mathematics with Applications

In this paper, we propose and analyze a first-order and a second-order time-stepping schemes for the anisotropic phase-field dendritic crystal growth model. The proposed schemes are based on an auxiliary variable approach for the Allen-Cahn equation and delicate treatment of the terms coupling the Allen-Cahn equation and temperature equation. The idea of the former is to introduce suitable auxiliary variables to facilitate construction of high order stable schemes for a large class of gradient flows. We propose a new technique to treat the coupling terms involved in the crystal growth model, and introduce suitable stabilization terms to result in totally decoupled schemes, which satisfy a discrete energy law without affecting the convergence order. A delicate implementation demonstrates that the proposed schemes can be realized in a very efficient way. That is, it only requires solving four linear elliptic equations and a simple algebraic equation at each time step. A detailed comparison with existing schemes is given, and the advantage of the new schemes is emphasized. As far as we know this is the first second-order scheme that is totally decoupled, linear, unconditionally stable for the dendritic crystal growth model with variable mobility parameter

Computers & Fluids

Thermal phase change problems arise in a large number of applications. In this paper, we consider a phase field model instead of the classical Stefan model to describe phenomena, which may appear in some complex phase change problems such as dendritic crystal growth, phase transformations in metallic alloys, etc. Our aim is to propose efficient and accurate schemes for the model, which is the coupling of a heat transfer equation and a phase field equation. The schemes are constructed based on an auxiliary variable approach for the phase field equation and semi-implicit treatment for the heat transfer equation. The main novelty of the paper consists in: (i) construction of the efficient schemes, which only requires solving several second-order elliptic problems with constant coefficients; (ii) proof of the unconditional stability of the schemes; (iii) fast high order solver for the resulting equations at each time step. A series of numerical examples are presented to verify the theoretical claims and to illustrate the efficiency of our method. As far as we know, it seems this is the first attempt made for the thermal phase change model of this type

An iterative domain decomposition algorithm for the grad(div) operator

This paper describes an iterative solution technique for partial differential equations involving the grad(div) operator, based on a domain decomposition. Iterations are performed to solve the solution on the interface. We identify the transmission relationships through the interface. We relate the approach to a Steklov-Poincare operator, and we illustrate the performance of technique through some numerical experiments