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The Keller-Segel (KS) chemotaxis equation is a widely studied mathematical model for understanding the collective behavior of cells in response to chemical gradients. This paper investigates the direct discontinuous Galerkin method with interface correction (DDGIC) for one-dimensional and two-dimensional KS equations governing the cell density and chemoattractant concentration. We establish error estimates for the proposed scheme under suitable smoothness assumptions of the exact solutions. Numerical experiments are conducted to validate the theoretical results. We explore the impact of different coefficient settings in the numerical fluxes on the error of the DDGIC method on uniform and nonuniform meshes. Our findings reveal that the DDGIC method achieves optimal convergence rates with any admissible coefficients for polynomials of odd degrees, while the accuracy of the cell density is sensitive to the numerical flux coefficient in the chemoattractant concentration for polynomials of even degrees. These results hold regardless of whether the mesh is uniform or nonuniform. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the local discontinuous Galerkin (LDG) methods coupled with multistep implicit–explicit (IMEX) time discretization to solve one-dimensional and two-dimensional nonlinear Schrödinger equations. In this approach, the nonlinear terms are treated explicitly, while the linear terms are handled implicitly. By the skew symmetry property of LDG operators and the properties of Gauss–Radau projections, we obtain error estimates for the prime and auxiliary variables, as well as the estimate for the time difference of the prime variables. These results, together with a carefully chosen numerical initial condition, allow us to obtain the optimal error estimate in both space and time for the fully discrete scheme. Numerical experiments are performed to verify the accuracy and efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

The generalized polynomial chaos (gPC) method is one of the most popular method for uncertainty quantification. Being essentially a spectral approach, the gPC method exhibits the spectral convergence rate which heavily depends on the regularity of the solution in the random space. Many regularity studies have been made for stochastic elliptic and parabolic equations while regularities studies of stochastic hyperbolic equations has long been infeasible due to its intrinsic difficulties. In this paper, we investigate the impact of uncertainty on the time-dependent radiative transfer equation (RTE) with nonhomogeneous boundary conditions, which sits somewhere between hyperbolic and parabolic equations. We theoretically prove the a-priori bound of the solution, its continuity with respect to the scattering coefficient, and its regularity in the random space. These studies can serve as a building block in understanding the influence of uncertainties in the passage from hyperbolic to parabolic equations. Moreover, we vigorously justify the validity of the gPC expansion ansatz based on the regularity study. Then the stochastic Galerkin method of the gPC approach is employed to discretize the random variable. We further conduct a delicate analysis to show the exponential decay rate of the gPC coefficients and establish the error estimates of the stochastic Galerkin approximation for both one-dimensional and multi-dimensional random space cases. Numerical tests are presented to verify our analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate hyperbolic systems with random inputs based on generalized polynomial chaos (gPC) approximations, which is one of the most popular methods for uncertainty quantification (UQ) and can be implemented with either the stochastic Galerkin (SG) method or the stochastic collocation (SC) method. One of the challenges for solving stochastic hyperbolic systems with the SG method is that the resulting deterministic system may not be hyperbolic. The lack of hyperbolicity may lead to the ill-posedness of the problem and the instability of numerical simulations. The main objective of this paper is to show that by approximating the solution in the random space with the SG method in a pseudo-spectral way with suitable quadrature rules, the SG scheme can be written as a SC scheme on a set of specific nodes. The resulting collocation scheme preserves the hyperbolicity of the original hyperbolic system, and is more efficient to implement. On the other hand, entropy conditions play an essential role in the well-posedness of hyperbolic conservation laws. Thus we approximate the resulted collocation scheme in space by the entropy stable nodal discontinuous Galerkin (DG) method Chen and Shu (J. Comput. Phys. 345:427-461, 2017), where the entropy stability is guaranteed by high order summation-by-parts operators, entropy conservative fluxes and entropy stable fluxes. Numerical experiments are performed to validate the accuracy and effectiveness of the proposed numerical red schemes. [ABSTRACT FROM AUTHOR]

In this paper we study the stationary radiative transfer equation with random coefficients. Galerkin methods are applied, which use orthogonal polynomials associated with the probability distribution of the random variables as basis functions in the random space. Such algorithms have been widely used for kinetic equations with random inputs, however, the corresponding numerical analysis is rare. In this paper we establish regularity theorems describing the smoothness properties of the solution, and investigate the convergence rate of N-term truncated polynomials under the spectral method framework. Numerical tests are conducted to demonstrate our analytical results. [ABSTRACT FROM AUTHOR]

The discontinuous Galerkin (DG) transport scheme is becoming increasingly popular in the atmospheric modeling due to its distinguished features, such as high-order accuracy and high-parallel efficiency. Despite the great advantages, DG schemes may produce unphysical oscillations in approximating transport equations with discontinuous solution structures including strong shocks or sharp gradients. Nonlinear limiters need to be applied to suppress the undesirable oscillations and enhance the numerical stability. It is usually very difficult to design limiters to achieve both high-order accuracy and non-oscillatory properties and even more challenging for the cubed-sphere geometry. In this paper, a simple and efficient limiter based on the weighted essentially non-oscillatory (WENO) methodology is incorporated in the DG transport framework on the cubed sphere. The uniform high-order accuracy of the resulting scheme is maintained because of the high-order nature of WENO procedures. Unlike the classic WENO limiter, for which the wide halo region may significantly impede parallel efficiency, the simple limiter requires only the information from the nearest neighboring elements without degrading the inherent high-parallel efficiency of the DG scheme. A bound-preserving filter can be further coupled in the scheme that guarantees the highly desirable positivity-preserving property for the numerical solution. The resulting scheme is high-order accurate, non-oscillatory, and positivity preserving for solving transport equations based on the cubed-sphere geometry. Extensive numerical results for several benchmark spherical transport problems are provided to demonstrate good results, both in accuracy and in non-oscillatory performance. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]