17,636 research outputs found
ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing
We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the Hâ1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation
Reducing phase error in long numerical binary black hole evolutions with sixth order finite differencing
We describe a modification of a fourth-order accurate ``moving puncture''
evolution code, where by replacing spatial fourth-order accurate differencing
operators in the bulk of the grid by a specific choice of sixth-order accurate
stencils we gain significant improvements in accuracy. We illustrate the
performance of the modified algorithm with an equal-mass simulation covering
nine orbits.Comment: 13 pages, 6 figure
Multiadaptive Galerkin Methods for ODEs III: A Priori Error Estimates
The multiadaptive continuous/discontinuous Galerkin methods mcG(q) and mdG(q)
for the numerical solution of initial value problems for ordinary differential
equations are based on piecewise polynomial approximation of degree q on
partitions in time with time steps which may vary for different components of
the computed solution. In this paper, we prove general order a priori error
estimates for the mcG(q) and mdG(q) methods. To prove the error estimates, we
represent the error in terms of a discrete dual solution and the residual of an
interpolant of the exact solution. The estimates then follow from interpolation
estimates, together with stability estimates for the discrete dual solution
- âŚ