A new strategy for implementing mapped WENO method

The mapping is a useful technique for improving the performance of WENO method. However, for the existed mapped WENO methods, two weaknesses still need to be overcome. First, too much computational cost is used, the computational cost of mapped WENO method is around [Formula: see text] of the original WENO method. Second, on the nonuniform meshes, too much memory space is used. We can take the WENO-RM method on nonuniform rectangle meshes for example, which is one of the famous mapped WENO methods. Comparing to original WENO method, the WENO-RM needs to store [Formula: see text] additional parameters, where [Formula: see text] is the cell number. In order to overcome these weaknesses, we propose a new strategy for implementing the mapped WENO method. Different from the previous works, the new strategy is based on the average weight. The new strategy has following major advantages: first, it can give the solution which is as accurate as the original mapped WENO method; second, it can save the computational cost, because it only uses slightly more computational cost than the original WENO method; third, it can save the memory space because it only needs to store a little additional parameter compared to the original WENO method.

A predictor–corrector scheme for solving nonlinear fractional differential equations with uniform and nonuniform meshes

In this paper, we develop two algorithms for solving linear and nonlinear fractional differential equations involving Caputo derivative. For designing new predictor–corrector (PC) schemes, we select the mesh points based on the two equal-height and equal-area distribution. Furthermore, the error bounds of PC schemes with uniform and equidistributing meshes are obtained. Finally, examples are constructed for illustrating the obtained PC schemes with uniform and equidistributing meshes. A comparative study is also presented.

An implicit-explicit finite-difference lattice Boltzmann subgrid method on nonuniform meshes

In this paper, an implicit-explicit finite-difference lattice Boltzmann method with subgrid model on nonuniform meshes is proposed. The implicit-explicit Runge–Kutta scheme, which has good convergence rate, is used for the time discretization and a mixed difference scheme, which combines the upwind scheme with the central scheme, is adopted for the space discretization. Meanwhile, the standard Smagorinsky subgrid model is incorporated into the finite-difference lattice Boltzmann scheme. The effects of implicit-explicit Runge–Kutta scheme and nonuniform meshes of present lattice Boltzmann method are discussed through simulations of a two-dimensional lid-driven cavity flow on nonuniform meshes. Moreover, the comparison simulations of the present method and multiple relaxation time lattice Boltzmann subgrid method are conducted qualitatively and quantitatively.

3D-2D Deformable Image Registration Using Feature-Based Nonuniform Meshes

By using prior information of planning CT images and feature-based nonuniform meshes, this paper demonstrates that volumetric images can be efficiently registered with a very small portion of 2D projection images of a Cone-Beam Computed Tomography (CBCT) scan. After a density field is computed based on the extracted feature edges from planning CT images, nonuniform tetrahedral meshes will be automatically generated to better characterize the image features according to the density field; that is, finer meshes are generated for features. The displacement vector fields (DVFs) are specified at the mesh vertices to drive the deformation of original CT images. Digitally reconstructed radiographs (DRRs) of the deformed anatomy are generated and compared with corresponding 2D projections. DVFs are optimized to minimize the objective function including differences between DRRs and projections and the regularity. To further accelerate the above 3D-2D registration, a procedure to obtain good initial deformations by deforming the volume surface to match 2D body boundary on projections has been developed. This complete method is evaluated quantitatively by using several digital phantoms and data from head and neck cancer patients. The feature-based nonuniform meshing method leads to better results than either uniform orthogonal grid or uniform tetrahedral meshes.