A Triangular Spectral Element Method for the 2-D Viscous Burgers Equation

A triangular spectral element method is established for the two-dimensional viscous Burgers equation. In the spatial direction, a new type of mapping is applied. We splice the local basis functions on each triangle into a global basis function. The second-order Crank-Nicolson/ leap-frog (CNLF) method is used for discretization in the time direction. Due to the use of a quasi-interpolation operator, the nonlinear term can be handled conveniently. We give the fully discrete scheme of the method and the implementation process of the algorithm. Numerical examples verify the effectiveness of this method.

Cubic Hermite Finite Element Method for Nonlinear Black-Scholes Equation Governing European Options

A numerical algorithm for solving a generalized Black-Scholes partial differential equation, which arises in European option pricing considering transaction costs is developed. The Crank-Nicolson method is used to discretize in the temporal direction and the Hermite cubic interpolation method to discretize in the spatial direction. The efficiency and accuracy of the proposed method are tested numerically, and the results confirm the theoretical behaviour of the solutions, which is also found to be in good agreement with the exact solution.

Conformal geometry on a class of embedded hypersurfaces in spacetimes

In this work, we study various geometric properties of embedded spacelike hypersurfaces in $1+1+2$ decomposed spacetimes with a preferred spatial direction, denoted $e^{\mu}$, which are orthogonal to the fluid flow velocity of the spacetime and admit a proper conformal transformation. To ensure non-vanishing and positivity of the scalar curvature of the induced metric on the hypersurface, we impose that the scalar curvature of the conformal metric is non-negative and that the associated conformal factor $\varphi$ satisfies $\hat{\varphi}^2+2\hat{\hat{\varphi}}>0$, where \hat{\ast} denotes derivative along the preferred spatial direction. Firstly, it is demonstrated that such hypersurface is either of Einstein type or the spatial twist vanishes on them, and that the scalar curvature of the induced metric is constant. It is then proved that if the hypersurface is compact and of Einstein type and admits a proper conformal transformation, then these hypersurfaces must be isomorphic to the 3-sphere, where we make use of some well known results on Riemannian manifolds admitting conformal transformations. If the hypersurface is not of Einstein type and have nowhere vanishing sheet expansion, we show that this conclusion fails. However, with the additional conditions that the scalar curvatures of the induced metric and the conformal metric coincide, the associated conformal factor is strictly negative and the third and higher order derivatives of the conformal factor vanish, the conclusion that the hypersurface is isomorphic to the 3-sphere follows. Furthermore, additional results are obtained under the conditions that the scalar curvature of a metric conformal to the induced metric is also constant. Finally, we consider some of our results in the context of locally rotationally symmetric spacetimes and show that, if the hypersurfaces are compact and not of Einstein type, then under certain specified conditions the hypersurface is isomorphic to the 3-sphere, where we constructed explicit examples of several proper conformal Killing vector fields along $e^{\mu}$.

A stabilizer free spatial weak Galerkin finite element methods for time-dependent convection-diffusion equations

In this paper, we propose a stabilizer free spatial weak Galerkin (SFSWG) finite element method for solving time-dependent convection diffusion equations based on weak form Eq. (4). SFSWG method in spatial direction and Euler difference operator Eq. (37) in temporal direction are used. The main reason for using the SFSWG method is because of its simple formulation that makes this algorithm more efficient and its implementation easier. The optimal rates of convergence of 𝒪⁢(hk) and 𝒪⁢(hk+1) in a discrete H1 and L2 norms, respectively, are obtained under certain conditions if polynomial spaces (Pk⁢(K),Pk⁢(e),[Pj⁢(K)]2) are used in the SFSWG finite element method. Numerical experiments are performed to verify the effectiveness and accuracy of the SFSWG method.

Buckling Optimization of Variable Stiffness Composite Panels for Curvilinear Fibers and Grid Stiffeners

Automated Fiber Placement (AFP) machines can manufacture composite panels with curvilinear fibers. In this article, the critical buckling load of grid-stiffened curvilinear fiber composite panels is maximized using a genetic algorithm. The skin is composed of layers in which the fiber orientation varies along one spatial direction. The design variables are the fiber orientation of the panel for each layer and the stiffener layout. Manufacturing constraints in terms of maximum curvature allowable by the AFP machine are imposed for both skin and stiffener fibers. The effect of manufacturing-induced gaps in the laminates is also incorporated. The finite element method is used to perform the buckling analyses. The panels are subjected to in-plane compressive and shear loads under several boundary conditions. Optimization results show that the percentage difference in the buckling load between curvilinear and straight fiber panels depends on the load case and boundary conditions.

Parameter-Uniform Numerical Scheme for Singularly Perturbed Delay Parabolic Reaction Diffusion Equations with Integral Boundary Condition

Numerical computation for the class of singularly perturbed delay parabolic reaction diffusion equations with integral boundary condition has been considered. A parameter-uniform numerical method is constructed via the nonstandard finite difference method for the spatial direction, and the backward Euler method for the resulting system of initial value problems in temporal direction is used. The integral boundary condition is treated using numerical integration techniques. Maximum absolute errors and the rate of convergence for different values of perturbation parameter ε and mesh sizes are tabulated for two model examples. The proposed method is shown to be parameter-uniformly convergent.

Precise Phase Structure in Four-fermion Interaction Model on Torus

We investigate finite-size effects on chiral symmetry breaking in a four-fermion interaction model at a finite temperature and a chemical potential. Applying the imaginary time formalism, the thermal quantum field theory is constructed on an S1 in the imaginary time direction. In this paper, the finite-size effect is introduced by a compact S1 spatial direction with a U(1)-valued boundary condition. Thus, we study the model on a $\mathbb {R}^{D-2} \times S^{1} \times S^{1}$ torus. Phase diagrams are obtained by evaluating the local minima of the effective potential in the leading order of the 1/N expansion. From the grand potential, we calculate the particle number density and the pressure, then we illustrate the correspondence with the phase structure. We obtain a stable size for which the sign of the pressure flips from negative to positive as the size decreases. Furthermore, the finite chemical potential expands the parameter range that the stable size exists.

Accelerated nonstandard finite difference method for singularly perturbed Burger-Huxley equations

Objective The main purpose of this paper is to present an accelerated nonstandard finite difference method for solving the singularly perturbed Burger-Huxley equation in order to produce more accurate solutions. Results The quasilinearization technique is used to linearize the nonlinear term. A nonstandard methodology of Mickens procedure is used in the spatial direction and also within the first order temporal direction that construct the first-order finite difference approximation to solve the considered problem numerically. To accelerate the rate of convergence from first to second-order, the Richardson extrapolation technique is applied. Numerical experiments were conducted to support the theoretical results.

Generating a 3D optical needle array with prescribed characteristics

Unlike the general optical needle along the optical axis, we propose a method to generate a three-dimensional (3D) array formed by optical needles with prescribed length and polarization direction. Moreover, the geometric model of the created array can be specified. With the aid of antenna array pattern synthesis theory and time reversal technology, a virtual uniform line source (ULS) antenna array arranged regularly near the confocal region of two high numerical apertures objectives is employed to obtain the required illumination in the pupil plane for creating the desired focal fields. Numerical results demonstrate that there is a one-to-one correspondence between the focal field and the virtual ULS antenna array elements. The length and polarization direction of the optical needles depends on the length and spatial direction of the virtual ULS antenna. The peculiarities of the focal field array, such as the polarization, length, number, spatial position and array structure, can be customized according to application requirements. The created optical needle array can be used for such application as 3D synchronous particle acceleration and manipulation, 3D parallel fabrication.