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In this paper we analyze an algorithm which solves the point projection and the “inversion” problems for parametric curves and surfaces. It consists of a geometric second order iteration which converges faster than existing first order methods, and whose sensitivity to the choice of initial values is small. Applications include the ICP algorithm for shape registration.

This paper is devoted to developing the Il'in-Allen-Southwell (IAS) parameter-uniform difference scheme on uniform meshes for solving strongly coupled systems of singularly perturbed convection-diffusion equations whose solutions may display boundary and/or interior layers, where strong coupling means that the solution components in the system are coupled together mainly through their first derivatives. By decomposing the coefficient matrix of convection term into the Jordan canonical form, we first construct the IAS scheme for 1-D systems and then extend the scheme to 2-D systems by employing an alternating direction technique. The robustness of the developed IAS scheme is illustrated through a series of numerical examples, including the magnetohydrodynamic duct flow problem with a high Hartmann number. Numerical evidence indicates that the IAS scheme is formally second order accurate in the sense that it is second order convergent when the perturbation parameter $\varepsilon$ is not too small and when the $\varepsilon$ is sufficiently small, the scheme is first order convergent in the discrete maximum norm uniformly in $\varepsilon$.

We propose second order accurate discontinuous Galerkin (DG) schemes which satisfy a strict maximum principle for general nonlinear convection-diffusion equations on unstructured triangular meshes. Motivated by genuinely high order maximum-principle-satisfying DG schemes for hyperbolic conservation laws, we prove that under suitable time step restriction for forward Euler time stepping, for general nonlinear convection-diffusion equations, the same scaling limiter coupled with second order DG methods preserves the physical bounds indicated by the initial condition while maintaining uniform second order accuracy. Similar to the purely convection cases, the limiters are mass conservative and easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle. Following the idea in earlier work, we extend the schemes to two-dimensional convection-diffusion equations on triangular meshes. There are no geometric constraints on the mesh such as angle acuteness. Numerical results including incompressible Navier-Stokes equations are presented to validate and demonstrate the effectiveness of the numerical methods.

We consider the numerical solution of the time-fractional diffusion-wave equation on two-dimensional and three-dimensional unbounded spatial domains. Introduce an artificial boundary and find the exact and approximate artificial boundary conditions for the given problem, which lead to a bounded computational domain. Using the exact or approximating boundary conditions on the artificial boundary, the original problem is reduced to an initial-boundary-value problem on the bounded computational domain which is respectively equivalent to or approximates the original problem. Finite difference methods are used to solve the reduced problems on the bounded computational domain and the stability of these finite difference methods are proved. The numerical results demonstrate that the method given in this paper is effective and feasible.