An Adjoint-Free CNOP–4DVar Hybrid Method for Identifying Sensitive Areas in Targeted Observations: Method Formulation and Preliminary Evaluation

This paper proposes a hybrid method, called CNOP–4 DVar, for the identification of sensitive areas in targeted observations, which takes the advantages of both the conditional nonlinear optimal perturbation(CNOP) and four-dimensional variational assimilation(4 DVar) methods. The proposed CNOP–4 DVar method is capable of capturing the most sensitive initial perturbation(IP), which causes the greatest perturbation growth at the time of verification;it can also identify sensitive areas by evaluating their assimilation effects for eliminating the most sensitive IP. To alleviate the dependence of the CNOP–4 DVar method on the adjoint model, which is inherited from the adjoint-based approach, we utilized two adjointfree methods, NLS-CNOP and NLS-4 DVar, to solve the CNOP and 4 DVar sub-problems, respectively. A comprehensive performance evaluation for the proposed CNOP–4 DVar method and its comparison with the CNOP and CNOP–ensemble transform Kalman filter(ETKF) methods based on 10 000 observing system simulation experiments on the shallow-water equation model are also provided. The experimental results show that the proposed CNOP–4 DVar method performs better than the CNOP–ETKF method and substantially better than the CNOP method.

Dual‑Wind Discontinuous Galerkin Methods for Stationary Hamilton‑Jacobi Equations and Regularized Hamilton‑Jacobi Equations

This paper develops and analyzes a new family of dual-wind discontinuous Galerkin(DG)methods for stationary Hamilton-Jacobi equations and their vanishing viscosity regularizations.The new DG methods are designed using the DG fnite element discrete calculus framework of[17]that defnes discrete diferential operators to replace continuous differential operators when discretizing a partial diferential equation(PDE).The proposed methods,which are non-monotone,utilize a dual-winding methodology and a new skewsymmetric DG derivative operator that,when combined,eliminate the need for choosing indeterminable penalty constants.The relationship between these new methods and the local DG methods proposed in[38]for Hamilton-Jacobi equations as well as the generalized-monotone fnite diference methods proposed in[13]and corresponding DG methods proposed in[12]for fully nonlinear second order PDEs is also examined.Admissibility and stability are established for the proposed dual-wind DG methods.The stability results are shown to hold independent of the scaling of the stabilizer allowing for choices that go beyond the Godunov barrier for monotone schemes.Numerical experiments are provided to gauge the performance of the new methods.

考虑冻融界面变化的土壤水热耦合模型

土壤的冻结和融化是土壤内部的重要物理过程,其冻融界面位置的移动影响土壤水热特性以及陆面和大气之间的水分能量交换,从而对陆面水热过程产生重要影响.本研究将土壤冻结和融化问题归结为考虑水热耦合的多运动边界问题,利用局部自适应变网格法进行数值离散,发展了考虑冻结和融化界面位置的移动对水热过程影响的土壤水热耦合模型.该模型基于陆面模式分层结构的敏感性试验表明:它能同时连续地追踪多个冻融界面,克服了等温线法在同一土壤层不能同时模拟多个冻融界面的困难,比高分辨率情形下的计算效率提高数倍且计算稳定.利用站点观测对土壤冻融界面的位置、土壤温度和未冻水含量所进行的模拟验证,进一步表明了该模型的合理性以及应用于陆面过程模式的模拟潜力.

A POSTERIORI ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF THE CAHN-HILLIARD EQUATION AND THE HELE-SHAW FLOW

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut ＋ △（ε△Au-ε^-1f（u）） = 0. It is shown that the a posteriori error bounds depends on ε^-1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct at2 adaptive algorithm for computing the solution of the Cahn- Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.

FINITE ELEMENT METHODS FOR A BI-WAVE EQUATION MODELING D-WAVE SUPERCONDUCTORS

In this paper we develop two conforming finite element methods for a fourth order bi-wave equation arising as a simplified Ginzburg-Landau-type model for d-wave superconductors in absence of applied magnetic field. Unlike the biharmonic operator A2, the bi-wave operator □^2 is not an elliptic operator, so the energy space for the bi-wave equation is much larger than the energy space for the biharmonic equation. This then makes it possible to construct low order conforming finite elements for the bi-wave equation. However, the existence and construction of such finite elements strongly depends on the mesh. In the paper, we first characterize mesh conditions which allow and not allow construction of low order conforming finite elements for approximating the bi-wave equation. We then construct a cubic and a quartic conforming finite element. It is proved that both elements have the desired approximation properties, and give optimal order error estimates in the energy norm, suboptimal （and optimal in some cases） order error estimates in the H1 and L^2 norm. Finally, numerical experiments are presented to guage the efficiency of the proposed finite element methods and to validate the theoretical error bounds.

MIXED INTERIOR PENALTY DISCONTINUOUS GALERKIN METHODS FOR ONE-DIMENSIONAL FULLY NONLINEAR SECOND ORDER ELLIPTIC AND PARABOLIC EQUATIONS

This paper is concerned with developing accurate and efficient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations （PDEs）. In the paper we present a general framework for constructing high order interior penalty discontinuous Galerkin （IP-DG） methods for approximating viscosity solutions of these fully nonlinear PDEs. In order to capture discontinuities of the second order derivative uxx of the solution u, three independent functions p1,p2 and p3 are introduced to represent numerical derivatives using various one-sided limits. The proposed DG frame- work, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a nonlinear problem into a mostly linear system of equations where the nonlinearity has been modified to include multiple values of the second order derivative uxz. The proposed framework extends a companion finite difference framework developed by the authors in [9] and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes. In addition to the nonstandard mixed formulation setting, another main idea is to replace the fully nonlinear differential operator by a numerical operator which is consistent with the differential operator and satisfies certain monotonicity （called g-monotonicity） properties. To ensure such a g-monotonicity, the crux of the construction is to introduce the numerical moment, which plays a critical role in the proposed DG frame- work. The g-monotonicity gives the DG methods the ability to select the mathematically ＂correct＂ solution （i.e., the viscosity solution） among all possible solutions. Moreover, the g-monotonicity allows for the possible development of more efficient nonlinear solvers as the special nonlinearity of the algebraic systems can be explored to decouple the equations. This paper also presents and analyzes numerical results for several numerical test problems which are used to guage the accuracy and efficiency of the proposed DG methods.

STRONG CONVERGENCE OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR A CLASS OF SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH MULTIPLICATIVE NOISE

This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations(SPDEs)with multiplicative noise.The nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitz and the nonlinearity in the drift term is only assumed to satisfy a one-sided Lipschitz condition.These assumptions are the same ones as the cases where numerical methods for general nonlinear stochastic ordinary differential equations(SODEs)under“minimum assumptions”were studied.As a result,the semilinear SPDEs considered in this paper are a direct generalization of these nonlinear SODEs.There are several difficulties which need to be overcome for this generalization.First,obviously the spatial discretization,which does not appear in the SODE case,adds an extra layer of difficulty.It turns out a spatial discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness matrix.In this paper we use a finite element interpolation technique to discretize the nonlinear drift term.Second,in order to prove the strong convergence of the proposed fully discrete finite element method,stability estimates for higher order moments of the H1-seminorm of the numerical solution must be established,which are difficult and delicate.A judicious combination of the properties of the drift and diffusion terms and some nontrivial techniques is used in this paper to achieve the goal.Finally,stability estimates for the second and higher order moments of the L^(2)-norm of the numerical solution are also difficult to obtain due to the fact that the mass matrix may not be diagonally dominant.This is done by utilizing the interpolation theory and the higher moment estimates for the H1-seminorm of the numerical solution.After overcoming these difficulties,it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of convergence.Numerical experiment results are also presented to validate the theoretical results and to demonstrate the efficiency of the proposed numerical method.