Unsteady mixed flows in non uniform closed water pipes: a Full Kinetic Approach

We recall the Pressurized and Free Surface model constructed for the modeling of unsteady mixed flows in closed water pipes where transition points between the free surface and pressurized flow are treated as a free boundary associated to a discontinuity of the gradient of pressure. Then we present a numerical kinetic scheme for the computations of unsteady mixed flows in closed water pipes. This kinetic method that we call FKA for “Full Kinetic Approach” is an easy and mathematically elegant way to deal with multiple transition points when the changes of state between free surface and pressurized flow occur. We use two approaches namely the “ghost waves approach” and the “Full Kinetic Approach” to treat these transition points. We show that this kinetic numerical scheme has the following properties: it is wet area conservative, under a CFL condition it preserves the wet area positive, it treats “naturally” the flooding zones and most of all it is very easy to implement it. Finally numerical experiments versus laboratory experiments are presented and the scheme produces results that are in a very good agreement. We also present a numerical comparison with analytic solutions for free surface flows in non uniform pipes: the numerical scheme has a very good behavior. A code to code comparison for pressurized flows is also conducted and leads to a very good agreement. We also perform a numerical experiment when flooding and drying flows may occur and finally make a numerical study of the order of the kinetic method.     

On the numerical approximations of stiff convection–diffusion equations in a circle

Our aim in this article is to study the numerical solutions of singularly perturbed convection–diffusion problems in a circular domain and provide as well approximation schemes, error estimates and numerical simulations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via boundary layer analysis, the so-called boundary layer elements which absorb the boundary layer singularities. Using a $P_1$ classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical scheme in a quasi-uniform mesh.     

A Helly-type theorem for intersections of orthogonally starshaped sets in \mathbb R^d

Let $\mathcal K$ be a finite family of orthogonal polytopes in $\mathbb R^d$ such that, for every nonempty subfamily $\mathcal K^\prime $ of $\mathcal K, \cap \{K : K$ in $\mathcal K^\prime \}$ , if nonempty, is a finite union of boxes whose intersection graph is a tree. Assume that every $d + 1$ (not necessarily distinct) members of $\mathcal K$ meet in a (nonempty) staircase starshaped set. Then $S \equiv \cap \{ K : K$ in $\mathcal K\}$ is nonempty and staircase starshaped.     

A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation

In this paper we devise a first-order-in-time, second-order-in-space, convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation. The unconditional unique solvability, energy stability and \(\ell ^\infty (0, T; \ell ^4)\) stability of the scheme are established. Using the a-priori stabilities, we prove error estimates for our scheme, in both the \(\ell ^\infty (0, T; \ell ^2)\) and \(\ell ^\infty (0, T; \ell ^\infty )\) norms. The proofs of these estimates are notable for the fact that they do not require point-wise boundedness of the numerical solution, nor a global Lipschitz assumption or cut-off for the nonlinear term. The \(\ell ^2\) convergence proof requires no refinement path constraint, while the one involving the \(\ell ^\infty \) norm requires only a mild linear refinement constraint, \(s \le C h\) . The key estimates for the error analyses take full advantage of the unconditional \(\ell ^\infty (0, T; \ell ^4)\) stability of the numerical solution and an interpolation estimate of the form \(\left\| \phi \right\| _4 \le C \left\| \phi \right\| _2^\alpha \left\| \nabla _h\phi \right\| _2^{1-\alpha },\alpha = \frac{4-D}{4},D=1,2,3\) , which we establish for finite difference functions. We conclude the paper with some numerical tests that confirm our theoretical predictions.     

On a time-splitting method for a scalar conservation law with a multiplicative stochastic perturbation and numerical experiments

In this paper, we present a numerical scheme for a first-order hyperbolic equation of nonlinear type perturbed by a multiplicative noise. The problem is set in a bounded domain D of ${\mathbb{R}^{d}}$ and with homogeneous Dirichlet boundary condition. Using a time-splitting method, we are able to show the existence of an approximate solution. The result of convergence of such a sequence is based on the work of Bauzet–Vallet–Wittbold (J Funct Anal, 2013), where the authors used the concept of measure-valued solution and Kruzhkov’s entropy formulation to show the existence and uniqueness of the stochastic weak entropy solution. Then, we propose numerical experiments by applying this scheme to the stochastic Burgers’ equation in the one-dimensional case.     

Precise and fast computation of Jacobian elliptic functions by conditional duplication

We developed a new method to compute the cosine amplitude function, $c\equiv \mathrm{cn}(u|m)$ , by using its double argument formula. The accumulation of round-off errors is effectively suppressed by the introduction of a complementary variable, $b\equiv 1-c$ , and a conditional switch between the duplication of $b$ and $c$ . The sine and delta amplitude functions, $s \equiv \mathrm{sn}(u|m)$ and $d \equiv \mathrm{dn}(u|m)$ , are evaluated from thus computed $b$ or $c$ by means of their identity relations. The new method is sufficiently precise as its errors are less than a few machine epsilons. Also, it is significantly faster than the existing procedures. In case of single precision computation, it runs more than 50 times faster than Bulirsch’s sncndn based on the Gauss transformation and 2.7 times faster than our previous method based on the simultaneous duplication of $s,c$ and $d$ . The ratios change to 7.6 and 3.5 respectively in case of the double precision environment.     

Existence and stability of ground states for fully discrete approximations of the nonlinear Schrödinger equation

In this paper we study the long time behavior of a discrete approximation in time and space of the cubic nonlinear Schrödinger equation on the real line. More precisely, we consider a symplectic time splitting integrator applied to a discrete nonlinear Schrödinger equation with additional Dirichlet boundary conditions on a large interval. We give conditions ensuring the existence of a numerical ground state which is close in energy norm to the continuous ground state. Such result is valid under a CFL condition of the form $\tau h^{-2}\le C$ where $\tau $ and $h$ denote the time and space step size respectively. Furthermore we prove that if the initial datum is symmetric and close to the continuous ground state $\eta $ then the associated numerical solution remains close to the orbit of $\eta ,\Gamma =\cup _\alpha \{e^{i\alpha }\eta \}$ , for very long times.     

Approximation of the Laplace and Stokes operators with Dirichlet boundary conditions through volume penalization: a spectral viewpoint

We report the results of a study on the spectral properties of Laplace and Stokes operators modified with a volume penalization term designed to approximate Dirichlet conditions in the limit when a penalization parameter, \(\eta \) , tends to zero. The eigenvalues and eigenfunctions are determined either analytically or numerically as functions of \(\eta \) , both in the continuous case and after applying Fourier or finite difference discretization schemes. For fixed \(\eta \) , we find that only the part of the spectrum corresponding to eigenvalues \(\lambda \lesssim \eta ^{-1}\) approaches Dirichlet boundary conditions, while the remainder of the spectrum is made of uncontrolled, spurious wall modes. The penalization error for the controlled eigenfunctions is estimated as a function of \(\eta \) and \(\lambda \) . Surprisingly, in the Stokes case, we show that the eigenfunctions approximately satisfy, with a precision \(O(\eta )\) , Navier slip boundary conditions with slip length equal to \(\sqrt{\eta }\) . Moreover, for a given discretization, we show that there exists a value of \(\eta \) , corresponding to a balance between penalization and discretization errors, below which no further gain in precision is achieved. These results shed light on the behavior of volume penalization schemes when solving the Navier–Stokes equations, outline the limitations of the method, and give indications on how to choose the penalization parameter in practical cases.     

Generalized Strong Boundary Points and Boundaries of Families of Continuous Functions

If ${\mathcal{A}}$ is a family of continuous functions on a locally compact Hausdorff space X, a boundary for ${\mathcal{A}}$ is a subset ${B \subset X}$ such that every ${f \in \mathcal{A}}$ attains its maximum modulus on B. In this work we generalize the concept of strong boundary points for families of functions and show that the collection of these generalized strong boundary points is always a boundary for ${\mathcal{A}}$ . We give conditions under which all boundaries for ${\mathcal{A}}$ consist of generalized strong boundary points and under which these points coincide with classical strong boundary points. When ${\mathcal{A}}$ has sufficient algebraic structure it is proven that this construction provides a unique boundary for ${\mathcal{A}}$ consisting of boundary points, and we conclude by demonstrating how this approach provides an alternate technique for proving the existence of the Choquet and Shilov boundaries in certain function algebras.     

The Robin and Wentzell-Robin Laplacians on Lipschitz Domains

Let $\Omega\subset{\Bbb R}^N$ be a bounded domain with Lipschitz boundary. We prove in the first part that a realization of the Laplacian with Robin boundary conditions $\frac{\partial u}{\partial \nu}+\beta u=0$ on the boundary $\partial \Omega$ generates a holomorphic $C_0$ -semigroup of angle $\pi/2$ on $C(\overline{\Omega})$ if $0<\beta_0\le \beta\in L^{\infty}(\partial \Omega)$ . With the same assumption on $\Omega$ and assuming that $0\le\beta\in L^{\infty}(\partial \Omega)$ , we show in the second part that one can define a realization of the Laplacian on $C(\overline{\Omega})$ with Wentzell-Robin boundary conditions $\Delta u+\frac{\partial u}{\partial \nu}+\beta u=0$ on the boundary $\partial \Omega$ and this operator generates a $C_0$ -semigroup.     

Sets with finite {\mathbb H}-perimeter and controlled normal

In the Heisenberg group, we prove that the boundary of sets with finite ${\mathbb H}$ -perimeter and having a bound on the measure theoretic normal is an ${\mathbb H}$ -Lipschitz graph. Then we show that if the normal is, on the boundary, the restriction of a continuous mapping, then the boundary is an ${\mathbb H}$ -regular surface.     

Convergence analysis of a high-order Nyström integral-equation method for surface scattering problems

In this paper we present a convergence analysis for the Nyström method proposed in [J Comput Phys 169 (1):80–110, 2001] for the solution of the combined boundary integral equation formulations of sound-soft acoustic scattering problems in three-dimensional space. This fast and efficient scheme combines FFT techniques and a polar change of variables that cancels out the kernel singularity. We establish the stability of the algorithms in the $L^2$ norm and we derive convergence estimates in both the $L^2$ and $L^\infty $ norms. In particular, our analysis establishes theoretically the previously observed super-algebraic convergence of the method in cases in which the right-hand side is smooth.     

A variational difference scheme for the one-dimensional diffusion equation

In this paper we construct a homogeneous variational difference scheme for the diffusion equation assuming its coefficients to be bounded and measurable; the order of convergence of the scheme is O(h²). We consider the boundary value problem (1) $$\frac{d}{{dx}}\left( {K(x)\frac{{du}}{{dx}}} \right) - g(x)u = - \frac{{dF}}{{dx}},0< x< X$$ subject to the boundary conditions (2) $$u(0) = a,u(X) = b$$ .     

Numerical extremal solutions for a mixed problem with singular {\phi} -Laplacian

We are concerned with extremal solutions for the mixed boundary value problem $$-\left(r^{N-1}\phi(u')\right)' = r^{N-1} g(r, u), \quad u'(0) = 0 = u(R),$$ where ${g : [0, R] \times \mathbb{R} \to \mathbb{R}}$ is a continuous function and ${\phi : (-\eta, \eta) \to \mathbb{R}}$ is an increasing homeomorphism with ${\phi(0) = 0.}$ We prove the existence of minimal and maximal solutions in presence of well-ordered lower and upper solutions and develop a numerical algorithm for theirs approximation. Also, we provide numerical experiments confirming the theoretical aspects.     

Steady Periodic Water Waves with Unbounded Vorticity: Equivalent Formulations and Existence Results

In this paper we consider the steady water wave problem for waves that possess a merely \(L_r\) -integrable vorticity, with \(r\in (1,\infty )\) being arbitrary. We first establish the equivalence of the three formulations – the velocity formulation, the stream function formulation, and the height function formulation – in the setting of strong solutions, regardless of the value of \(r\) . Based upon this result and using a suitable notion of weak solution for the height function formulation, we then establish, by means of local bifurcation theory, the existence of small-amplitude capillary and capillary–gravity water waves with an \(L_r\) -integrable vorticity.     

Low-rank approximation of integral operators by using the Green formula and quadrature

Approximating integral operators by a standard Galerkin discretisation typically leads to dense matrices. To avoid the quadratic complexity it takes to compute and store a dense matrix, several approaches have been introduced including $\mathcal {H}$ -matrices. The kernel function is approximated by a separable function, this leads to a low rank matrix. Interpolation is a robust and popular scheme, but requires us to interpolate in each spatial dimension, which leads to a complexity of $m^d$ for $m$ -th order. Instead of interpolation we propose using quadrature on the kernel function represented with Green’s formula. Due to the fact that we are integrating only over the boundary, we save one spatial dimension compared to the interpolation method and get a complexity of $m^{d-1}$ .     

The gluing formula of the refined analytic torsion for an acyclic Hermitian connection

In the previous study by Huang and Lee (arXiv:1004.1753) we introduced the well-posed boundary conditions ${{\mathcal P}_{-, {\mathcal L}_{0}}}$ and ${{\mathcal P}_{+, {\mathcal L}_{1}}}$ for the odd signature operator to define the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the gluing formula of the refined analytic torsion for an acyclic Hermitian connection with respect to the boundary conditions ${{\mathcal P}_{-, {\mathcal L}_{0}}}$ and ${{\mathcal P}_{+, {\mathcal L}_{1}}}$ . In this case the refined analytic torsion consists of the Ray-Singer analytic torsion, the eta invariant and the values of the zeta functions at zero. We first compare the Ray-Singer analytic torsion and eta invariant subject to the boundary condition ${{\mathcal P}_{-, {\mathcal L}_{0}}}$ or ${{\mathcal P}_{+, {\mathcal L}_{1}}}$ with the Ray-Singer analytic torsion subject to the relative (or absolute) boundary condition and eta invariant subject to the APS boundary condition on a compact manifold with boundary. Using these results together with the well known gluing formula of the Ray-Singer analytic torsion subject to the relative and absolute boundary conditions and eta invariant subject to the APS boundary condition, we obtain the main result.     

Identification of a convolution kernel in a control problem for the heat equation with a boundary memory term

We consider the evolution of the temperature \(u\) in a material with thermal memory characterized by a time-dependent convolution kernel \(h\) . The material occupies a bounded region \(\Omega \) with a feedback device controlling the external temperature located on the boundary \(\Gamma \) . Assuming both \(u\) and \(h\) unknown, we formulate an inverse control problem for an integrodifferential equation with a nonlinear and nonlocal boundary condition. Existence and uniqueness results of a solution to the inverse problem are proved.     

Suitable families of boxes and kernels of staircase starshaped sets in {\mathbb{R}^d}

Let S be an orthogonal polytope in ${\mathbb{R}^d}$ . There exists a suitable family ${\mathcal{C}}$ of boxes with ${S = \cup \{C : C {\rm in} \mathcal{C}\}}$ such that the following properties hold:

The staircase kernel Ker S is a union of boxes in ${\mathcal{C}}$ . Let ${\mathcal{V}}$ be the family of vertices of boxes in ${\mathcal{C}}$ , and let ${v_o\, \epsilon \mathcal{V}}$ . Point v _o belongs to Ker S if and only if v _o sees via staircase paths in S every point w in ${\mathcal{V}}$ . Moreover, these staircase paths may be selected to consist of edges of boxes in ${\mathcal{C}}$ . Let B be a box in ${\mathcal{C}}$ with vertices of B in Ker S. Box B lies in Ker S if and only if, for some b in rel int B and for every translate H of a coordinate hyperplane at ${b, b \epsilon}$ Ker (H ∩ S). For point p in S, p belongs to Ker S if and only if, for every x in S, there exist some p ? x geodesic λ (p, x) and some corresponding ${\mathcal{C}}$ - chain D containing λ (p, x) such that D is staircase starshaped at p.