SINGULAR SOLUTIONS FOR A CONVECTION DIFFUSION EQUATION WITH ABSORPTION

In this paper we discuss the existence and nonexistence of singular solutions for a porous medium equations with convection and absorption terms.

Mixed time discontinuous space-time finite element method for convection diffusion equations

A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method.

THE POINTWISE ESTIMATES OF SOLUTIONS FOR A NONLINEAR CONVECTION DIFFUSION REACTION EQUATION

This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the optimal Lp,1≤ p ≤ ＋∞,convergence rate of solutions for small initial data.Then we establish the local existence of solutions,the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data.Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.

STREAMLINE DIFFUSION F.E.M. FOR SOBOLEV EQUATIONS WITH CONVECTION DOMINATED TERM

In this paper,a streamline diffusion F.E.M. for linear Sobolev equations with convection dominated term is given.According to the range of space time F.E mesh parameter h ,two choices for artifical diffusion parameter δ are presented,and for the corresponding computation schemes the stability and error estimates in suitable norms are estabilished.

Meshfree Finite Volume Element Method for Constrained Optimal Control Problem Governed by Random Convection Diffusion Equations

In this paper,we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random coefficients.There are two contributions of this paper.Firstly,we establish a scheme to approximate the optimality system by using the finite volume element method in the physical space and the meshfree method in the probability space,which is competitive for high-dimensional random inputs.Secondly,the a priori error estimates are derived for the state,the co-state and the control variables.Some numerical tests are carried out to confirm the theoretical results and demonstrate the efficiency of the proposed method.

GLOBAL EXISTENCE AND L^p ESTIMATES FOR SOLUTIONS OF DAMPED WAVE EQUATION WITH NONLINEAR CONVECTION IN MULTI-DIMENSIONAL SPACE

In this article, the author studies the Cauchy problem of the damped wave equation with a nonlinear convection term in multi-dimensions. The author shows that a classical solution to the Cauchy problem exists globally in time under smallness condition on the initial perturbation. Furthermore, the author obtains the L^p （2 ≤ p ≤ ∞） decay estimates of the solution.

THE SOLUTIONS OF STEADY－STATE CONVECTION EQUATIONS IN THE SPACES THAT POSSESS RESTORI NGNUCLEUS

In this paper ,in the space that possesses restoring nucleus, we obtain analyticsolutions in the series form for the steady-state convection diffusion equation The solutions have the following characteristics: (1) they ave given in the accurate form:(2)they can be calculated in the explicit way, without solving the eguations;(3) the error of the approximate solution will be monotonically decreased under the meaning of the norm of the spaces when a cardinal term is added in the procedure of numerical solution .Finally, we calculated the example in [2] the result shows that our solution is more accurate than that in [2].

THE STABILITY AND CONVERGENCE OF THE FINITE ANALYTIC METHOD FOR THE NUMERICAL SOLUTION OF CONVECTIVE DIFFUSION EQUATION

In this paper we make a close study of the finite analytic method by means of the maximum principles in differential equations and give the proof of the stability and convergence of the finite analytic method.

A simplified two-dimensional boundary element method with arbitrary uniform mean flow

To reduce computational costs, an improved form of the frequency domain boundary element method（BEM） is proposed for two-dimensional radiation and propagation acoustic problems in a subsonic uniform flow with arbitrary orientation. The boundary integral equation（BIE） representation solves the two-dimensional convected Helmholtz equation（CHE） and its fundamental solution, which must satisfy a new Sommerfeld radiation condition（SRC） in the physical space. In order to facilitate conventional formulations, the variables of the advanced form are expressed only in terms of the acoustic pressure as well as its normal and tangential derivatives, and their multiplication operators are based on the convected Green＇s kernel and its modified derivative. The proposed approach significantly reduces the CPU times of classical computational codes for modeling acoustic domains with arbitrary mean flow. It is validated by a comparison with the analytical solutions for the sound radiation problems of monopole,dipole and quadrupole sources in the presence of a subsonic uniform flow with arbitrary orientation.

DIFFERENCE SCHEME AND NUMERICAL SIMULATION BASED ON MIXED FINITE ELEMENT METHOD FOR NATURAL CONVECTION PROBLEM

The non_stationary natural convection problem is studied. A lowest order finite difference scheme based on mixed finite element method for non_stationary natural convection problem, by the spatial variations discreted with finite element method and time with finite difference scheme was derived, where the numerical solution of velocity, pressure, and temperature can be found together, and a numerical example to simulate the close square cavity is given, which is of practical importance.

EXPONENTIAL TIME DIFFERENCING-PADE FINITE ELEMENT METHOD FOR NONLINEAR CONVECTION-DIFFUSION-REACTION EQUATIONS WITH TIME CONSTANT DELAY

In this paper,ETD3-Padéand ETD4-PadéGalerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions.An ETD-based RK is used for time integration of the corresponding equation.To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator,the Padéapproach is used for such an exponential operator approximation,which in turn leads to the corresponding ETD-Padéschemes.An unconditional L^(2) numerical stability is proved for the proposed numerical schemes,under a global Lipshitz continuity assumption.In addition,optimal rate error estimates are provided,which gives the convergence order of O(k^(3)+h^(r))(ETD3-Padé)or O(k^(4)+h^(r))(ETD4-Padé)in the L^(2)norm,respectively.Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.

VARIATIONAL DISCRETIZATION FOR OPTIMAL CONTROL GOVERNED BY CONVECTION DOMINATED DIFFUSION EQUATIONS

In this paper, we study variational discretization for the constrained optimal control problem governed by convection dominated diffusion equations, where the state equation is approximated by the edge stabilization Galerkin method. A priori error estimates are derived for the state, the adjoint state and the control. Moreover, residual type a posteriori error estimates in the L^2-norm are obtained. Finally, two numerical experiments are presented to illustrate the theoretical results.

MULTIPLICATIVE SCHWARZ ALGORITHM WITH TIME STEPPING ALONG CHARACTERISTIC FOR CONVECTION DIFFUSION EQUATIONS

Examines the convection diffusion problems using domain decomposition method. Presentation of continuous and discrete convection diffusion equations; Kinds of multiplicative Schwarz algorithms; Optimal order error estimate results.

An exponential expanding meshes sequence and finite difference method adopted for two-dimensional elliptic equations

We demonstrate a new nonuniform mesh finite difference method to obtain accurate solutions for the elliptic partial differential equations in two dimensions with nonlinear first-order partial derivative terms.The method will be based on a geometric grid network area and included among the most stable differencing scheme in which the nine-point spatial finite differences are implemented,thus arriving at a compact formulation.In general,a third order of accuracy has been achieved and a fourth-order truncation error in the solution values will follow as a particular case.The efficiency of using geometric mesh ratio parameter has been shown with the help of illustrations.The convergence of the scheme has been established using the matrix analysis,and irreducibility is proved with the help of strongly connected characteristics of the iteration matrix.The difference scheme has been applied to test convection diffusion equation,steady state Burger’s equation,ocean model and a semi-linear elliptic equation.The computational results confirm the theoretical order and accuracy of the method.

Numerical Resolution Near t=0 of Nonlinear Evolution Equations in the Presence of Corner Singularities in Space Dimension 1

The incompatibilities between the initial and boundary data will cause singularities at the time-space corners,which in turn adversely affect the accuracy of the numerical schemes used to compute the solutions.We study the corner singularity issue for nonlinear evolution equations in 1D,and propose two remedy procedures that effectively recover much of the accuracy of the numerical scheme in use.Applications of the remedy procedures to the 1D viscous Burgers equation,and to the 1D nonlinear reaction-diffusion equation are presented.The remedy procedures are applicable to other nonlinear diffusion equations as well.

L^(p) Decay Rate for a Nonlinear Convection Diffusion Reaction Equation in R~n

This paper studies the asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in R^(n).Firstly,the global existence and uniqueness of classical solutions for small initial data are established.Then,we obtain the L^(p),2≤p≤+∞decay rate of solutions.The approach is based on detailed analysis of the Green function of the linearized equation with the technique of long wave-short wave decomposition and the Fourier analysis.

Investigating and Mitigating Failure Modes in Physics-Informed Neural Networks(PINNs)

This paper explores the difficulties in solving partial differential equations(PDEs)using physics-informed neural networks(PINNs).PINNs use physics as a regularization term in the objective function.However,a drawback of this approach is the requirement for manual hyperparameter tuning,making it impractical in the absence of validation data or prior knowledge of the solution.Our investigations of the loss landscapes and backpropagated gradients in the presence of physics reveal that existing methods produce non-convex loss landscapes that are hard to navigate.Our findings demonstrate that high-order PDEs contaminate backpropagated gradients and hinder convergence.To address these challenges,we introduce a novel method that bypasses the calculation of high-order derivative operators and mitigates the contamination of backpropagated gradients.Consequently,we reduce the dimension of the search space and make learning PDEs with non-smooth solutions feasible.Our method also provides a mechanism to focus on complex regions of the domain.Besides,we present a dual unconstrained formulation based on Lagrange multiplier method to enforce equality constraints on the model’s prediction,with adaptive and independent learning rates inspired by adaptive subgradient methods.We apply our approach to solve various linear and non-linear PDEs.

Comparison of Different Order Adams-Bashforth Methods in an Atmospheric General Circulation Model

The Asselin-Robert time As an attractive alternative filter used in the leaptYog scheme does degrade the accuracy of calculations. to leapfrog time differencing, the second-order Adams-Bashforth method is not subject to time splitting instability and keeps excellent calculation accuracy. A second-order Adams- Bashforth model has been developed, which represents better stability, excellent convergence and improved simulation of prognostic variables. Based on these results, the higher-order Adams-Bashforth methods are developed on the basis of NCAR （National Center for Atmospheric Research） CAM 3.1 （Community Atmosphere Model 3.1） and the characteristics of dynamical cores are analyzed in this paper. By using Lorenz nonlinear convective equations, the filtered leapfrog scheme shows an excellent pattern for eliminating 2At wave solutions after 20 steps but represents less computational solution accuracy. The fourth-order Adams- Bashforth method is closely converged to the exact solution and provides a reference against which other methods may be compared. Thus, the Adams-Bashforth methods produce more accurate and convergent solution with differencing order increasing. The Held-Suarez idealized test is carried out to demonstrate that all methods have similar climate states to the results of many other global models for long-term integration. Besides, higher-order methods perform better in mass conservation and exhibit improvement in simulating tropospheric westerly jets, which is likely equivalent to the advantages of increasing horizontal resolutions. Based on the idealized baroclinic wave test, a better capability of the higher-order method in maintaining simulation stability is convinced. Furthermore, after the baroclinic wave is triggered through overlaying the steady-state initial conditions with the zonal perturbation, the higher-order method has a better ability in the simulation of baroclinic wave perturbation.

SUPPLEMENT AND IMPROVEMENT OF HOLLY-PREISSMANN SCHEME

Using the undetermined coefficient method, Holly-Preissmann scheme isimproved effectively. The scheme with the minus velocity is added, and a new conservative scheme isalso presented on the basis of original scheme. The simulations of the new scheme accord with theexact result, which enhances its applicability in the engineering.

A Parallel Algorithm for the Convection Diffusion Problem

Based on the second-order compact upwind scheme, a group explicit method for solving the two-dimensional time-independent convection-dominated diffusion problem is developed. The stability of the group explicit method is proven strictly. The method has second-order accuracy and good stability. This explicit scheme can be used to solve all Reynolds number convection-dominated diffusion problems. A numerical test using a parallel computer shows high efficiency. The numerical results conform closely to the analytic solution.