Numerical approximation for singular second order differential equations

We consider numerical approximation of solutions of singular second order differential equations. In particular, we study the backward (or implicit) Euler method. We prove results concerning consistency, global error and stability. We show that the global error is linear with respect to the step size. Numerical results are also given, which demonstrate the linear convergence and we compare the numerical results with known approximations.     

Unconditional stability of a partitioned IMEX method for magnetohydrodynamic flows

Magnetohydrodynamic (MHD) flows are governed by Navier–Stokes equations coupled with Maxwell equations through coupling terms. We prove the unconditional stability of a partitioned method for the evolutionary full MHD equations, at high magnetic Reynolds number, written in the Elsässer variables. The method we analyze is a first-order one-step scheme, which consists of implicit discretization of the subproblem terms and explicit discretization of the coupling terms.     

A partitioned second‐order method for magnetohydrodynamic flows at small magnetic reynolds numbers

This article aims to study the partitioned method for magnetohydrodynamic flows at small magnetic Reynolds numbers. We design a partitioned second‐order method and show that this method is stable under a time step () restrict condition. Our method can decouple the magnetohydrodynamic equations so that we can solve two relatively simple subproblems separately at each time step, which is computationally economic. A complete theoretical analysis of error estimates is also given. Finally, we present numerical experiments to support our theory.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1966–1986, 2017     

On a simplified model for radiating flows

We consider a simplified model arising in radiation hydrodynamics which is based on the barotropic Navier–Stokes system describing the macroscopic fluid motion and a P1-approximation (see below) of the transport equation modeling the propagation of radiative intensity. We establish global-in-time existence of strong solutions for the associated Cauchy problem when initial data are close to a stable radiative equilibrium and local existence for large data with no vacuum. All our results are stated in the so-called critical Besov spaces.     

Large eddy simulation for turbulent magnetohydrodynamic flows

We investigate the mathematical properties of a model for the simulation of large eddies in turbulent, electrically conducting, viscous, incompressible flows. We prove existence and uniqueness of solutions for the simplest (zeroth) closed MHD model (1.7), we show that its solutions converge to the solution of the MHD equations as the averaging radii converge to zero, and derive a bound on the modeling error. Furthermore, we show that the model preserves the properties of the 3D MHD equations: the kinetic energy and the magnetic helicity are conserved, while the cross helicity is approximately conserved and converges to the cross helicity of the MHD equations, and the model is proven to preserve the Alfvén waves, with the velocity converging to that of the MHD, as δ₁,δ₂ tend to zero. We perform computational tests that verify the accuracy of the method and compare the conserved quantities of the model to those of the averaged MHD.     

Numerical solutions for system of second order boundary value problems

We investigate some numerical methods for computing approximate solutions of a system of second order boundary value problems associated with obstacle, unilateral and contact problems. We show that cubic spline method gives approximations which are better than that computed by higher order spline and finite difference techniques.     

Convergence analysis of some first order and second order time accurate gradient schemes for semilinear second order hyperbolic equations

This work is devoted to the convergence analysis of finite volume schemes for a model of semilinear second order hyperbolic equations. The model includes for instance the so‐called Sine‐Gordon equation which appears for instance in Solid Physics (cf. Fang and Li, Adv Math (China) 42 (2013), 441–457; Liu et al., Numer Methods Partial Differ Equ 31 (2015), 670–690). We are motivated by two works. The first one is Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043) where a recent class of nonconforming finite volume meshes is introduced. The second one is Eymard et al. (Numer Math 82 (1999), 91–116) where a convergence of a finite volume scheme for semilinear elliptic equations is provided. The mesh considered in Eymard et al. (Numer Math 82 (1999), 91–116) is admissible in the sense of Eymard et al. (Elsevier, Amsterdam, 2000, 723–1020) and a convergence of a family of approximate solutions toward an exact solution when the mesh size tends to zero is proved. This article is also a continuation of our previous two works (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1278–1321; Bradji, Numer Methods Partial Differ Equ 29 (2013), 1–39) which dealt with the convergence analysis of implicit finite volume schemes for the wave equation. We use as discretization in space the generic spatial mesh introduced in Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043), whereas the discretization in time is performed using a uniform mesh. Two finite volume schemes are derived using the discrete gradient of Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043). The unknowns of these two schemes are the values at the center of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. The first scheme is inspired from the previous work (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1–39), whereas the second one (in which the discretization in time is performed using a Newmark method) is inspired from the work (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1278–1321). Under the assumption that the mesh size of the time discretization is small, we prove the existence and uniqueness of the discrete solutions. If we assume in addition to this that the exact solution is smooth, we derive and prove three error estimates for each scheme. The first error estimate is concerning an estimate for the error between a discrete gradient of the approximate solution and the gradient of the exact solution whereas the second and the third ones are concerning the estimate for the error between the exact solution and the discrete solution in the discrete seminorm of and in the norm of . The convergence rate is proved to be for the first scheme and for the second scheme, where (resp. k) is the mesh size of the spatial (resp. time) discretization. The existence, uniqueness, and convergence results stated above do not require any relation between k and . The analysis presented in this work is also applicable in the gradient schemes framework. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 5–33, 2017     

Mathematical analysis of a simplified Hookean dumbbells model arising from viscoelastic flows

A stochastic model corresponding to a simplified Hookean dumbbells viscoelastic fluid is considered, the convective terms being disregarded. Existence on a fixed time interval is proved provided the data are small enough, using the implicit function theorem and a maximum regularity property for a three fields Stokes problem.     

On the stability of jetlike magnetohydrodynamic flows

The linear stability problem is under study for steady axisymmetric translational flows of a density-homogeneous nonviscous incompressible ideal conducting fluid with free surface and “frozen-in” poloidal magnetic field. By the direct Lyapunov method, some sufficient conditions are obtained for the stability of these flows under small long-wave perturbations with the same symmetry. These stability conditions have partial converses; and, for unstable stationary flows, an a priori exponential lower bound is constructed on the growth of small perturbations under consideration, while the increment of the appearing exponent serves as an arbitrary positive parameter. An illustrative analytical example is given of steady flows with superimposed small long-wave axisymmetric perturbations growing in time in accordance with the estimate.     

Numerical differentiation for the second order derivatives of functions of two variables

A regularized optimization problem for computing numerical differentiation for the second order derivatives of functions with two variables from noisy values at scattered points is discussed in this article. We prove the existence and uniqueness of the solution to this problem, provide a constructive scheme for the solution which is based on bi-harmonic Green's function and give a convergence estimate of the regularized solution to the exact solution for the problem under a simple choice of regularization parameter. The efficiency of the constructive scheme is shown by some numerical examples.     

A new spectral-homotopy analysis method for solving a nonlinear second order BVP

A modification of the homotopy analysis method (HAM) for solving nonlinear second-order boundary value problems (BVPs) is proposed. The implementation of the new approach is demonstrated by solving the Darcy–Brinkman–Forchheimer equation for steady fully developed fluid flow in a horizontal channel filled with a porous medium. The model equation is solved concurrently using the standard HAM approach and numerically using a shooting method based on the fourth order Runge–Kutta scheme. The results demonstrate that the new spectral homotopy analysis method is more efficient and converges faster than the standard homotopy analysis method.     

New exact solutions for magnetohydrodynamic flows of an Oldroyd-B fluid

This paper presents the new exact analytical solutions for magnetohydrodynamic (MHD) flows of an Oldroyd-B fluid. The explicit expressions for the velocity field and the associated tangential stress are established by using the Laplace transform method. Three characteristic examples: (i) flow due to impulsive motion of plate, (ii) flow due to uniformly accelerated plate, and (iii) flow due to non-uniformly accelerated plate are considered. The solutions for the hydrodynamic flows are special cases of the presented solutions. Moreover, the similar solutions corresponding to Maxwell and Newtonian fluids in the presence as well as absence of a magnetic field appear as the limiting cases of our solutions. The influences of the exerted magnetic field on the flow are also graphically presented and discussed. In particular, graphical results for the Oldroyd-B fluid are compared with those of a Newtonian fluid.     

Development of a high order discretization scheme for solving fully nonlinear magnetohydrodynamic equations

We have developed fully fourth order accurate compact finite difference discretization scheme for the Navier-Stokes equations coupled with Maxwell''s equations. The implementation is done in cylindrical polar geometry. Due to the full-MHD modeling of physical flow, the modeled equations are fully nonlinear coupled hydrodynamic equations which are again coupled with Maxwells equations. In our computations, we have accounted for the induced magnetic field in the flow of an electrically conducting fluid in an external magnetic field. The code is tested against available experimental and theoretical data where applicable. It is observed that a smaller grid of $64 \times 64$ is sufficient for weakly nonlinear problems and higher grids up to $512 \times 512$ are needed as the degree of nonlinearities grow in the modeled equation. In the absence of magnetic field, a discontinuity of total drag coefficient and separation length is noted for $Re=73$ which is in agreement with literature. When the magnetic Reynolds number $Rm<1$ separation length decreases linearly with strength of magnetic field on a log-log scale whereas if $Rm>1$, it decreases nonlinearly, at a much faster rate. Thermal boundary layer thickness decreases as the strength of magnetic field increases and it forces the thermal convection to take place in a laminar structure as observed from thermal contour lines. Finally, using divided differences, we establish that the accuracy of the proposed numerical scheme is in fact fourth order.     

Numerical analysis of a high order method for state-dependent delay integral equations

In this paper we study the piecewise collocation method for a class of functional integral equations with state-dependent delays that is, where the delays depend on the solution. It is well known that these equations typically have discontinuity in the solution or its derivatives at the initial point of integration domain. This discontinuity propagated along the integration interval giving rise to subsequent points, called ”singular points”, which can not be determined priori and the solution derivatives in these points are smoothed out along the interval. Most of the known numerical methods for this type of equations are generally very sensitive to the singular points and therefore must have a process that detects these points and insert them into the mesh to guarantee the required accuracy. Here, we present a numerical algorithm based on the piecewise collocation method and an approach for tracking the singular points relying on the recent strategy for implicit delay differential equations which has been proposed by Guglielmi and Hairer in 2008. The convergence analysis of the method is investigated and some numerical experiments are presented for clarifying the robustness of the method.     

Construction and analysis of a new second order nonconforming finite element

构造一个新的二阶非协调有限元,通过新的技巧和方法,将该单元用于一般二阶椭圆问题,得到插值误差和相容性误差,同时达到O(h2)误差阶.     

Existence and uniqueness of solutions for the magnetohydrodynamic flow of a second grade fluid

In this work, we consider the flow of a second grade fluid in a conducting domain of and in the presence of a magnetic field. When the initial data are of arbitrary size, we prove that the solution of the magnetohydrodynamics problem exists for a small time and is unique. We also show the global existence of solutions for small initial data. Copyright © 2012 John Wiley & Sons, Ltd.     

A full second order variational model for multiscale texture analysis

We present a second order image decomposition model to perform denoising and texture extraction. We look for the decomposition f=u+v+w where u is a first order term, v a second order term and w the (0 order) remainder term. For highly textured images the model gives a two-scale texture decomposition: u can be viewed as a macro-texture (larger scale) whose oscillations are not too large and w is the micro-texture (very oscillating) that may contain noise. We perform mathematical analysis of the model and give numerical examples.     

Complex analysis in subsystems of second order arithmetic

This research is motivated by the program of Reverse Mathematics. We investigate basic part of complex analysis within some weak subsystems of second order arithmetic, in order to determine what kind of set existence axioms are needed to prove theorems of basic analysis. We are especially concerned with Cauchy’s integral theorem. We show that a weak version of Cauchy’s integral theorem is proved in RCAo. Using this, we can prove that holomorphic functions are analytic in RCAo. On the other hand, we show that a full version of Cauchy’s integral theorem cannot be proved in RCAo but is equivalent to weak König’s lemma over RCAo.     

Numerical analysis for generalized Forchheimer flows of slightly compressible fluids

In this article, we will consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for density. The long‐time numerical approximation of the nonlinear degenerate parabolic equation with time dependent boundary conditions is studied. The stability for all time is established in a continuous time scheme and a discrete backward Euler scheme. A Gronwall's inequality‐type is used to study the asymptotic behavior of the solution. Error estimates for the solution are derived for both continuous and discrete time procedures. Numerical experiments confirm the theoretical analysis regarding convergence rates.