A new class of semi-implicit methods with linear complexity for nonlinear fractional differential equations

We propose a new class of semi-implicit methods for solving nonlinear fractional differential equations and study their stability. Several versions of our new schemes are proved to be unconditionally stable by choosing suitable parameters. Subsequently, we develop an efficient strategy to calculate the discrete convolution for the approximation of the fractional operator in the semi-implicit method and we derive an error bound of the fast convolution. The memory requirement and computational cost of the present semi-implicit methods with a fast convolution are about $O(N\log n_T)$ and $O(Nn_T\log n_T)$, respectively, where $N$ is a suitable positive integer and $n_T$ is the final number of time steps. Numerical simulations, including the solution of a system of two nonlinear fractional diffusion equations with different fractional orders in two-dimensions, are presented to verify the effectiveness of the semi-implicit methods.Comment: 25 pages, 10 figure

Embedded Implicit Stand-ins for Animated Meshes: a Case of Hybrid Modelling

In this paper we address shape modelling problems, encountered in computer animation and computer games development that are difficult to solve just using polygonal meshes. Our approach is based on a hybrid modelling concept that combines polygonal meshes with implicit surfaces. A hybrid model consists of an animated polygonal mesh and an approximation of this mesh by a convolution surface stand-in that is embedded within it or is attached to it. The motions of both objects are synchronised using a rigging skeleton. This approach is used to model the interaction between an animated mesh object and a viscoelastic substance, normally modelled in implicit form. The adhesive behaviour of the viscous object is modelled using geometric blending operations on the corresponding implicit surfaces. Another application of this approach is the creation of metamorphosing implicit surface parts that are attached to an animated mesh. A prototype implementation of the proposed approach and several examples of modelling and animation with near real-time preview times are presented

Immersed Boundary Smooth Extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods

The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet it only achieves first-order spatial accuracy near embedded boundaries. In this paper, we introduce a new high-order numerical method which we call the Immersed Boundary Smooth Extension (IBSE) method. The IBSE method achieves high-order accuracy by smoothly extending the unknown solution of the PDE from a given smooth domain to a larger computational domain, enabling the use of simple Cartesian-grid discretizations (e.g. Fourier spectral methods). The method preserves much of the flexibility and robustness of the original IB method. In particular, it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta-functions to communicate information between the computational grid and the boundary. We present a fast algorithm for solving elliptic equations, which forms the basis for simple, high-order implicit-time methods for parabolic PDE and implicit-explicit methods for related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat, Burgers', and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise convergence for Dirichlet problems and third-order pointwise convergence for Neumann problems