A Nonlinear Representation of Model Uncertainty in a Convective-Scale Ensemble Prediction System

How to accurately address model uncertainties with consideration of the rapid nonlinear error growth characteristics in a convection-allowing system is a crucial issue for performing convection-scale ensemble forecasts.In this study,a new nonlinear model perturbation technique for convective-scale ensemble forecasts is developed to consider a nonlinear representation of model errors in the Global and Regional Assimilation and Prediction Enhanced System(GRAPES)Convection-Allowing Ensemble Prediction System(CAEPS).The nonlinear forcing singular vector(NFSV)approach,that is,conditional nonlinear optimal perturbation-forcing(CNOP-F),is applied in this study,to construct a nonlinear model perturbation method for GRAPES-CAEPS.Three experiments are performed:One of them is the CTL experiment,without adding any model perturbation;the other two are NFSV-perturbed experiments,which are perturbed by NFSV with two different groups of constraint radii to test the sensitivity of the perturbation magnitude constraint.Verification results show that the NFSV-perturbed experiments achieve an overall improvement and produce more skillful forecasts compared to the CTL experiment,which indicates that the nonlinear NFSV-perturbed method can be used as an effective model perturbation method for convection-scale ensemble forecasts.Additionally,the NFSV-L experiment with large perturbation constraints generally performs better than the NFSV-S experiment with small perturbation constraints in the verification for upper-air and surface weather variables.But for precipitation verification,the NFSV-S experiment performs better in forecasts for light precipitation,and the NFSV-L experiment performs better in forecasts for heavier precipitation,indicating that for different precipitation events,the perturbation magnitude constraint must be carefully selected.All the findings above lay a foundation for the design of nonlinear model perturbation methods for future CAEPSs.

Numerical solutions of singular integral equations for planar rectangular interfacial crack in three dimensional bimaterials

Stress intensity factors for a three dimensional rectangular interfacial crack were considered using the body force method. In the numerical calculations, unknown body force densities were approximated by the products of the fundamental densities and power series; here the fundamental densities are chosen to express singular stress fields due to an interface crack exactly. The calculation shows that the numerical results are satisfied. The stress intensity factors for a rectangular interface crack were indicated accurately with the varying aspect ratio, and bimaterial parameter.

Stability of Navier–Stokes System with Singular External Force in Fourier–Herz Space

We prove the asymptotic properties of the solutions to the 3D Navier–Stokes system with singular external force, by making use of Fourier localization method, the Littlewood–Paley theory and some subtle estimates in Fourier–Herz space. The main idea of the proof is motivated by that of Cannone et al. [J. Differential Equations, 314, 316–339(2022)]. We deal either with the nonstationary problem or with the stationary problem where solution may be singular due to singular external force. In this paper, the Fourier–Herz space includes the function space of pseudomeasure type used in Cannone et al. [J. Differential Equations, 314, 316–339(2022)]

A Simple 3D Immersed InterfaceMethod for Stokes Flow with Singular Forces on Staggered Grids

In this paper,a fairly simple 3D immersed interface method based on the CG-Uzawa type method and the level set representation of the interface is employed for solving three-dimensional Stokes flow with singular forces along the interface.The method is to apply the Taylor’s expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes.A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy.The Stokes equations are discretized involving the correction terms on staggered grids and then solved by the conjugate gradient Uzawa type method.The major advantages of the present method are the special simplicity,the ability in handling the Dirichlet boundary conditions,and no need of the pressure boundary condition.The method can also preserve the volume conservation and the discrete divergence free condition very well.The numerical results show that the proposed method is second order accurate and efficient.

An Implementation ofMAC Grid-Based IIM-Stokes Solver for Incompressible Two-Phase Flows

Abstract.In this paper,a novel implementation of immersed interface method combined with Stokes solver on a MAC staggered grid for solving the steady two-fluid Stokes equations with interfaces.The velocity components along the interface are introduced as two augmented variables and the resulting augmented equation is then solved by the GMRES method.The augmented variables and/or the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity,and are interpolated using cubic splines and are then applied to the fluid through the jump conditions.The Stokes equations are discretized on a staggered Cartesian grid via a second order finite difference method and solved by the conjugate gradient Uzawa-typemethod.The numerical results show that the overall scheme is second order accurate.The major advantages of the present IIM-Stokes solver are the efficiency and flexibility in terms of types of fluid flow and different boundary conditions.The proposed method avoids solution of the pressure Poisson equation,and comparisons are made to show the advantages of time savings by the present method.The generalized two-phase Stokes solver with correction terms has also been applied to incompressible two-phase Navier-Stokes flow.

The Immersed Interface Method for Simulating Two-Fluid Flows

We develop the immersed interface method(IIM)to simulate a two-fluid flow of two immiscible fluids with different density and viscosity.Due to the surface tension and the discontinuous fluid properties,the two-fluid flow has nonsmooth velocity and discontinuous pressure across the moving sharp interface separating the two fluids.The IIM computes the flow on a fixed Cartesian grid by incorporating into numerical schemes the necessary jump conditions induced by the interface.We present how to compute these necessary jump conditions from the analytical principal jump conditions derived in[Xu,DCDS,Supplement 2009,pp.838-845].We test our method on some canonical two-fluid flows.The results demonstrate that the method can handle large density and viscosity ratios,is second-order accurate in the infinity norm,and conserves mass inside a closed interface.

An Immersed Interface Method for the Simulation of Inextensible Interfaces in Viscous Fluids

In this paper,an immersed interface method is presented to simulate the dynamics of inextensible interfaces in an incompressible flow.The tension is introduced as an augmented variable to satisfy the constraint of interface inextensibility,and the resulting augmented system is solved by the GMRES method.In this work,the arclength of the interface is locally and globally conserved as the enclosed region undergoes deformation.The forces at the interface are calculated from the configuration of the interface and the computed augmented variable,and then applied to the fluid through the related jump conditions.The governing equations are discretized on a MAC grid via a second-order finite difference scheme which incorporates jump contributions and solved by the conjugate gradient Uzawa-type method.The proposed method is applied to several examples including the deformation of a liquid capsule with inextensible interfaces in a shear flow.Numerical results reveal that both the area enclosed by interface and arclength of interface are conserved well simultaneously.These provide further evidence on the capability of the present method to simulate incompressible flows involving inextensible interfaces.