John Paul Roop

Numerical Approximation of a Time Dependent, Nonlinear, Space-Fractional Diffusion Equation

In this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a nonlocal quadratic nonlinearity. The analysis is performed for a general fractional order diffusion operator. The nonlinear term studied is a product of the unknown function and a convolution operator of order 0. Convergence of the approximation and a priori error estimates are given. Numerical computations are included, which confirm the theoretical predictions.

A Two-Level Discretization Method for the Smagorinsky Model

A two-level method for discretizing the Smagorinsky model for the numerical simulation of turbulent flows is proposed and analyzed. In the two-level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single-step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two-level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We demonstrate numerically that, for an appropriate choice of grids, the two-level algorithm is significantly more efficient than the standard one-level algorithm. We also provide a rigorous numerical analysis of the two-level method, which yields appropriate scalings between the coarse and fine mesh-sizes (H and h, respectively), and the radius of the spatial filter used in the Smagorinsky model (

A Bounded Artificial Viscosity Large Eddy Simulation Model

\delta

Regularity of the solution to 1-D fractional order diffusion equations

). Our analysis provides a mathematical answer to the large eddy simulation mystery of finding the scaling between the filter radius and numerical mesh-size.

Multilevel TV Denoising Methods for Anisotropic Tensor Product Wavelets

In this paper, we present a rigorous numerical analysis for a bounded artificial viscosity model (

Instructional Selection of Active Learning and Traditional Courses Increases Student Achievement in College Mathematics.

\tau=\mu\delta^\sigma a(\delta\|\nabla^s\bm{u}\|_F)\nabla^s\bm{u})

2-Dimensional Geometric Transforms for Edge Detection

for the numerical simulation of turbulent flows. In practice, the commonly used Smagorinsky model (

Variational formulation for the stationary fractional advection dispersion equation

\tau=(c_s\delta)^2\|\nabla^s\bm{u}\|_F\,\nabla^s\bm{u}

Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2

) is overly dissipative and yields unphysical results. To date, several methods for “clipping” the Smagorinsky viscosity have proven useful in improving the physical characteristics of the simulated flow. However, such heuristic strategies strongly rely upon a priori knowledge of the flow regime. The bounded artificial viscosity model relies on a highly nonlinear, but monotone and smooth, semilinear elliptic form for the artificial viscosity. For this model, we have introduced a variational computational strategy, provided finite element error convergence estimates, and included several computational examples indicating its improvement on the overly diffusive Smagorinsky model.

Variational solution of fractional advection dispersion equations on bounded domains in ℝd

In this article we investigate the solution of the steady-state fractional diffusion equation on a bounded domain in