In this paper, we develop third-order conservative sign-preserving and steady-state-preserving time integrations and seek their applications in multispecies and multireaction chemical reactive flows. In this problem, the density and pressure are nonnegative, and the mass fraction for the ith species, denoted as zi, 1<=i<=M, should be between 0 and 1, where M is the total number of species. There are four main difficulties in constructing high-order bound-preserving techniques for multispecies and multireaction detonations. First of all, most of the bound-preserving techniques available are based on Euler forward time integration. Therefore, for problems with stiff source, the time step will be significantly limited. Secondly, the mass fraction does not satisfy a maximum-principle and hence it is not easy to preserve the upper bound 1. Thirdly, in most of the previous works for gaseous denotation, the algorithm relies on second-order Strang splitting methods where the ux and stiff source terms can be solved separately, and the extension to high-order time discretization seems to be complicated. Finally, most of the previous ODE solvers for stiff problems cannot preserve the total mass and the positivity of the numerical approximations at the same time. In this paper, we will construct third-order conservative sign-preserving Rugne-Kutta and multistep methods to overcome all these difficulties. The time integrations do not depend on the Strang splitting, i.e. we do not split the flux and the stiff source terms. Moreover, the time discretization can handle the stiff source with large time step and preserves the steady-state. Numerical experiments will be given to demonstrate the good performance of the bound-preserving technique and the stability of the scheme for problems with stiff source terms.

Poisson equation on point cloud with Dirichlet boundary condition plays important role in many problems. In this paper, we use the volume constraint proposed by Du et.al to handle the Dirichlet boundary condition in the point integral method for Poisson equation on point cloud. We prove that the solution given by volume constraint converges to the true solution as the point cloud converges to the underlying smooth manifold.

he incompressibility constraint makes Navier-Stokes equations difficult. A reformulation to a better posed problem is needed before solving it numerically. The sequential regularization method (SRM) is a reformulation which combines the penalty method with a stabilization method in the context of constrained dynamical systems and has the benefit of both methods. In the paper, we study the existence and uniqueness for the solution of the SRM and provide a simple proof of the convergence of the solution of the SRM to the solution of the Navier-Stokes equations. We also give error estimates for the time discretized SRM formulation.

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Consider the image restoration from incomplete noisy frequency data with total variation and sparsity regularizing penalty terms. Firstly, we establish an unconstrained optimization model with dierent smooth approximations on the regularizing terms. Then, to weaken the amount of computations for cost functional with total variation term, the alternating iterative scheme is developed to obtain the exact solution through shrinkage thresholding in inner loop, while the nonlinear Euler equation is appropriately linearized at each iteration in exterior loop, yielding a linear system with diagonal coefficient matrix in frequency domain. Finally the linearized iteration is proven to be convergent in generalized sense for suitable regularizing parameters, and the error between the linearized iterative solution and the one gotten from the exact nonlinear Euler equation is rigorously estimated, revealing the essence of the proposed alternative iteration scheme. Numerical tests for different configurations show the validity of the proposed scheme, compared with some existing algorithms.