In 1-equation URANS models of turbulence, the eddy viscosity is given by $\nu_T = 0.55l(x, t)\sqrt{k(x, t)}.$ The length scale $l$ must be pre-specified and $k(x, t)$ is determined by solving a nonlinear partial differential equation. We show that in interesting cases the spacial mean of $k(x, t)$ satisfies a simple ordinary differential equation. Using its solution in $\nu_T$ results in a 1/2-equation model. This model has attractive analytic properties. Further, in comparative tests in 2d and 3d the velocity statistics produced by the 1/2-equation model are comparable to those of the full 1-equation model.

In this paper, we establish a novel reduced-order weak Galerkin (ROWG) finite element method for solving parabolic equation with nonlinear compression coefficient. We first present the classical weak Galerkin finite element discretization scheme and derive the optimal error estimates. Then we apply a proper orthogonal decomposition (POD) technique to develop the ROWG method, which can effectively reduce degrees of freedom and CPU time. The optimal order error estimates are also derived, and the algorithm flow is provided. Finally, some numerical experiments illustrate the performance of the ROWG method. The numerical results show that the proposed ROWG method is efficient for solving nonlinear parabolic equations.

In this article, for solving fluid-fluid interaction problem, we consider a Navier-Stokes-$ω$/Navier-Stokes-$ω$ model, which includes two Navier-Stokes-$ω$ equations coupled by some nonlinear interface conditions. Based on an auxiliary variable, we propose a fully discrete, decouple finite element scheme. We adopt the backward Euler scheme and mixed finite element approximation for temporal-spatial discretization, and explicit treatment for the interface terms and nonlinear terms. Moreover, the proposed scheme is shown to be unconditionally stable. Then, we establish error estimate of the numerical solution. Finally, with a series of numerical experiments we illustrate the stability and effectiveness of the proposed scheme and its ability to capture basic phenomenological features of the fluid-fluid interaction.

This article introduces the hybridizable discontinuous Galerkin (HDG) approach to numerically approximate the solution of a linear hyperbolic integro-differential equation. A priori error estimates for semi-discrete and fully discrete schemes are developed. It is shown that the optimal order of convergence is achieved for both scalar and flux variables. To achieve that, an intermediate projection is introduced for the semi-discrete error analysis, and it also shows that this projection achieves convergence of order $h^{k+3/2}$ for $k ≥ 1.$ Next, superconvergence is achieved for the scalar variable using element-by-element post-processing. For the fully discrete error analysis, the central difference scheme and the mid-point rule approximate the derivative and the integral term, respectively. Hence, the second order of convergence is achieved in the temporal direction. Finally, numerical experiments have been performed to validate the theory developed in this article.

A class of particular interface problems, which is derived from Bazant-Storey-Kornyshev (BSK) theory to account for the electrostatic correlation in concentrated electrolytes, is studied in this paper. It involves a modified fourth-order Poisson-Fermi equation in solvents and a second-order Poisson equation in solutes with high-contrast coefficients, where nonhomogeneous interface conditions are introduced over the interface that divides solutes from solvents. A type of interface-fitted finite element method is developed and analyzed for this interface problem, and optimal error estimates are obtained for all variables in both $H^1$ and $L^2$ norms. Numerical experiments validate all attained theoretical results through two mathematical examples, as well as the electrostatic correlation phenomenon in concentrated electrolytes through a physical example, practically, where the electrostatic stress and interactional forces in the concentrated electrolyte are computed to reveal the charge reversal phenomenon that is governed by the BSK theory.

In this paper, we propose an energy-conservative finite difference time domain (FDTD) method for solving the coupled nonlinear Klein-Gordon equations (CNKGEs) in the nonrelativistic limit regime, involving a small parameter $0 < ε ≪ 1$ which is inversely proportional to the speed of light. Employing cut-off technique, we analyze rigorously error estimates for the numerical method. Numerical results are reported to confirm the energy-conservative property and the error results in $l^2$ norm and $H^1$ norm under different values of $ε.$

Fourier analysis works well for the finite difference schemes of the linear partial differential equations. However, the presence of nonlinear terms leads to the fact that the method cannot be applied directly to deal with nonlinear problems. In the current work, we introduce an effective approach to enable Fourier methods to effectively deal with nonlinear problems and elaborate on it in detail by rigorously proving that the difference scheme for two-dimensional nonlinear problem considered in this paper is strictly unconditionally stable and convergent. Further, some numerical experiments are performed to confirm the rates of convergence and the robustness of the numerical scheme.