A new Semi-Lagrangian scheme is proposed to discretize the surface convection-diffusion equation. The other involved equations including the the level-set convection equation, the re-initialization equation and the extension equation are also solved by S-L schemes. The S-L method removes both the CFL condition and the stiffness caused by the surface Laplacian, allowing larger time step than the Eulerian method. The method is extended to the block-structured adaptive mesh. Numerical examples are given to demonstrate the efficiency of the S-L method.

We analyse the performance of a recursive Monte Carlo method for the Bayesian estimation of the static parameters of a discrete-time state-space Markov model. The algorithm employs two layers of particle filters to approximate the posterior probability distribution of the model parameters. In particular, the first layer yields an empirical distribution of samples on the parameter space, while the filters in the second layer are auxiliary devices to approximate the (analytically intractable) likelihood of the parameters. This approach relates the novel algorithm to the recent sequential Monte Carlo square method, which provides a nonrecursive solution to the same problem. In this paper we investigate the approximation of integrals of real bounded functions with respect to the posterior distribution of the system parameters. Under assumptions related to the compactness of the parameter support and the stability and continuity of the sequence of posterior distributions for the state-space model, we prove that the Lp norms of the approximation errors vanish asymptotically (as the number of Monte Carlo samples generated by the algorithm increases) and uniformly over time. We also prove that, under the same assumptions, the proposed scheme can asymptotically identify the parameter values for a class of models. We conclude the paper with a numerical example that illustrates the uniform convergence results by exploring the accuracy and stability of the proposed algorithm operating with long sequences of observations.

In this paper, we study the numerical solution of singularly perturbed time-dependent convection-diffusion problems. To solve these problems, the backward Euler method is first applied to discretize the time derivative on a uniform mesh, and the classical upwind finite difference scheme is used to approximate the spatial derivative on an arbitrary nonuniform grid. Then, in order to obtain an adaptive grid for all temporal levels, we construct a positive monitor function, which is similar to the arc-length monitor function. Furthermore, the ε-uniform convergence of the fully discrete scheme is derived for the numerical solution. Finally, some numerical results are given to support our theoretical results.

The bubble packing method can generate high-quality node sets in simple and complex domains. However, its efficiency remains to be improved. This study is a part of an ongoing effort to introduce several acceleration schemes to reduce the cost of simulation. Firstly, allow the viscosity coefficient c in the bubble governing equations to change according the coordinate of the bubble which are defined separately as odd and normal bubbles, and meanwhile with the saw-shape relationship with time or iterations. Then, in order to relieve the over crowded initial bubble placement, two coefficients w1 and w2 are introduced to modify the insertion criterion. The range of those two coefficients are discussed to be w1 = 1, w2 ∈ [0.5,0.8]. Finally, a self-adaptive termination condition is logically set when the stable system equilibrium is achieved. Numerical examples illustrate that the computing cost can significantly decrease by roughly 80% via adopting various combination of proper schemes (except the uniform placement example), and the average qualities of corresponding Delaunay triangulation substantially exceed 0.9. It shows that those strategies are efficient and can generate a node set with high quality.

A novel method for boundary constrained tetrahedral mesh generation is proposed based on Advancing Front Technique (AFT) and conforming Delaunay triangulation. Given a triangulated surface mesh, AFT is firstly applied to mesh several layers of elements adjacent to the boundary. The rest of the domain is then meshed by the conforming Delaunay triangulation. The non-conformal interface between two parts of meshes are adjusted. Mesh refinement and mesh optimization are then preformed to obtain a more reasonable-sized mesh with better quality. Robustness and quality of the proposed method is shown. Convergence proof of each stage as well as the whole algorithm is provided. Various numerical examples are included as well as the quality of the meshes.