This paper introduces a formulation of the variable density incompressible Navier-Stokes equations by modifying the nonlinear terms in a consistent way. For Galerkin discretizations, the formulation leads to full discrete conservation of mass, squared density, momentum, angular momentum and kinetic energy without the divergence-free constraint being strongly enforced. In addition to favorable conservation properties, the formulation is shown to make the density field invariant to global shifts. The effect of viscous regularizations on conservation properties is also investigated. Numerical tests validate the theory developed in this work. The new formulation shows superior performance compared to other formulations from the literature, both in terms of accuracy for smooth problems and in terms of robustness.
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This paper is the second in a series of works on weak convergence of one-step schemes for solving stochastic differential equations (SDEs) with one-sided Lipschitz conditions. It is known that the super-linear coefficients may lead to a blowup of moments of solutions and numerical solutions and thus affect the convergence of numerical methods. Wang et al. (2023, IMA J. Numer. Anal.) have analyzed weak convergence of one-step numerical schemes when solutions to SDEs have all finite moments. Therein some modified Euler schemes have been discussed about their weak convergence orders. In this work, we explore the effects of limited orders of moments on the weak convergence of a family of explicit schemes. The schemes are based on approximations/modifications of terms in the Ito-Talyor expansion. We provide a systematic but simple way to establish weak convergence orders for these schemes. We present several numerical examples of these schemes and show their weak convergence orders.
For high-order accurate schemes such as discontinuous Galerkin (DG) methods solving Fokker-Planck equations, it is desired to efficiently enforce positivity without losing conservation and high-order accuracy, especially for implicit time discretizations. We consider an optimization-based positivity-preserving limiter for enforcing positivity of cell averages of DG solutions in a semi-implicit time discretization scheme, so that the point values can be easily enforced to be positive by a simple scaling limiter on the DG polynomial in each cell. The optimization can be efficiently solved by a first-order splitting method with nearly optimal parameters, which has an $\mathcal{O}(N)$ computational complexity and is flexible for parallel computation. Numerical tests are shown on some representative examples to demonstrate the performance of the proposed method.
Discontinuous Galerkin (DG) methods for solving elliptic equations are gaining popularity in the computational physics community for their high-order spectral convergence and their potential for parallelization on computing clusters. However, problems in numerical relativity with extremely stretched grids, such as initial data problems for binary black holes that impose boundary conditions at large distances from the black holes, have proven challenging for DG methods. To alleviate this problem we have developed a primal DG scheme that is generically applicable to a large class of elliptic equations, including problems on curved and extremely stretched grids. The DG scheme accommodates two widely used initial data formulations in numerical relativity, namely the puncture formulation and the extended conformal thin-sandwich (XCTS) formulation. We find that our DG scheme is able to stretch the grid by a factor of $\sim 10^9$ and hence allows to impose boundary conditions at large distances. The scheme converges exponentially with resolution both for the smooth XCTS problem and for the nonsmooth puncture problem. With this method we are able to generate high-quality initial data for binary black hole problems using a parallelizable DG scheme. The code is publicly available in the open-source SpECTRE numerical relativity code.
This paper introduces a nonconforming virtual element method for general second-order elliptic problems with variable coefficients on domains with curved boundaries and curved internal interfaces. We prove arbitrary order optimal convergence in the energy and $L^2$ norms, confirmed by numerical experiments on a set of polygonal meshes. The accuracy of the numerical approximation provided by the method is shown to be comparable with the theoretical analysis.
This paper presents a convolution tensor decomposition based model reduction method for solving the Allen-Cahn equation. The Allen-Cahn equation is usually used to characterize phase separation or the motion of anti-phase boundaries in materials. Its solution is time-consuming when high-resolution meshes and large time scale integration are involved. To resolve these issues, the convolution tensor decomposition method is developed, in conjunction with a stabilized semi-implicit scheme for time integration. The development enables a powerful computational framework for high-resolution solutions of Allen-Cahn problems, and allows the use of relatively large time increments for time integration without violating the discrete energy law. To further improve the efficiency and robustness of the method, an adaptive algorithm is also proposed. Numerical examples have confirmed the efficiency of the method in both 2D and 3D problems. Orders-of-magnitude speedups were obtained with the method for high-resolution problems, compared to the finite element method. The proposed computational framework opens numerous opportunities for simulating complex microstructure formation in materials on large-volume high-resolution meshes at a deeply reduced computational cost.
We present a novel spatial discretization for the Cahn-Hilliard equation including transport. The method is given by a mixed discretization for the two elliptic operators, with the phase field and chemical potential discretized in discontinuous Galerkin spaces, and two auxiliary flux variables discretized in a divergence-conforming space. This allows for the use of an upwind-stabilized discretization for the transport term, while still ensuring a consistent treatment of structural properties including mass conservation and energy dissipation. Further, we couple the novel spatial discretization to an adaptive time stepping method in view of the Cahn-Hilliard equation's distinct slow and fast time scale dynamics. The resulting implicit stages are solved with a robust preconditioning strategy, which is derived for our novel spatial discretization based on an existing one for continuous Galerkin based discretizations. Our overall scheme's accuracy, robustness, efficient time adaptivity as well as structure preservation and stability with respect to advection dominated scenarios are demonstrated in a series of numerical tests.
In this paper we consider the numerical solution of the two-dimensional time-dependent partial integro-differential complementarity problem (PIDCP) that holds for the value of American-style options under the two-asset Kou jump-diffusion model. Following the method of lines (MOL), we derive an efficient numerical method for the pertinent PIDCP. Here, for the discretization of the nonlocal double integral term, an extension is employed of the fast algorithm by Toivanen (2008) in the case of the one-asset Kou jump-diffusion model. For the temporal discretization, we study a useful family of second-order diagonally implicit Runge-Kutta (DIRK) methods. Their adaptation to the semidiscrete two-dimensional Kou PIDCP is obtained by means of an effective iteration introduced by d'Halluin, Forsyth & Labahn (2004) and d'Halluin, Forsyth & Vetzal (2005). Ample numerical experiments are presented showing that the proposed numerical method achieves a favourable, second-order convergence behaviour to the American two-asset option value as well as to its Greeks Delta and Gamma.
The incompressible Euler equations are an important model system in computational fluid dynamics. Fast high-order methods for the solution of this time-dependent system of partial differential equations are of particular interest: due to their exponential convergence in the polynomial degree they can make efficient use of computational resources. To address this challenge we describe a novel timestepping method which combines a hybridised Discontinuous Galerkin method for the spatial discretisation with IMEX timestepping schemes, thus achieving high-order accuracy in both space and time. The computational bottleneck is the solution of a (block-) sparse linear system to compute updates to pressure and velocity at each stage of the IMEX integrator. Following Chorin's projection approach, this update of the velocity and pressure fields is split into two stages. As a result, the hybridised equation for the implicit pressure-velocity problem is reduced to the well-known system which arises in hybridised mixed formulations of the Poisson- or diffusion problem and for which efficient multigrid preconditioners have been developed. Splitting errors can be reduced systematically by embedding this update into a preconditioned Richardson iteration. The accuracy and efficiency of the new method is demonstrated numerically for two time-dependent testcases that have been previously studied in the literature.
This paper focuses on decoupled finite element methods for the fourth-order exterior differential equation. Based on differential complexes and the Helmholtz decomposition, the fourth-order exterior differential equation is decomposed into two second-order exterior differential equations and one generalized Stokes equation. A family of conforming finite element methods are developed for the decoupled formulation. Numerical results are provided for verifying the decoupled finite element methods of the biharmonic equation in three dimensions.
This paper proposes a new generalized linear model with fractional binomial distribution. Zero-inflated Poisson/negative binomial distributions are used for count data that has many zeros. To analyze the association of such a count variable with covariates, zero-inflated Poisson/negative binomial regression models are widely used. In this work, we develop a regression model with the fractional binomial distribution that can serve as an additional tool for modeling the count response variable with covariates. Data analysis results show that on some occasions, our model outperforms the existing zero-inflated regression models.
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