Multigrid schemes for high order finite difference methods on summation-by-parts form are studied by comparing the effect of different interpolation operators. By using the standard linear prolongation and restriction operators, the Galerkin condition leads to inaccurate coarse grid discretizations. In this paper, an alternative class of interpolation operators that bypass this issue and preserve the summation-by-parts property on each grid level is considered. Clear improvements of the convergence rate for relevant model problems are achieved.

Multigrid schemes for high order finite difference methods on summation-by-parts form are studied by comparing the effect of different interpolation operators. By using the standard linear prolongation and restriction operators, the Galerkin condition leads to inaccurate coarse grid discretizations. In this paper, an alternative class of interpolation operators that bypass this issue and preserve the summation-by-parts property on each grid level is considered. Clear improvements of the convergence rate for relevant model problems are achieved.

In this paper we consider one-dimensional flow governed by Burgers' equation. We analyze two iterative methods for data assimilation problem for this equation. One of them so called adjoint optimization method, is based on minimization in L ²-norm. We show that this minimization problem is ill-posed but the adjoint optimization iterative method is regularizing, and represents the well-known Landweber method in inverse problems. The second method is based on L ²-minimization of the gradient. We prove that this problem always has a solution. We present numerical comparisons of these two methods.

The viscous Burgers' equation with nonlinear viscosity is considered. The equation is written as a quasilinear parabolic equation in divergence form, and existence of a weak solution is shown. The proof is based on Galerkin approximations which converges in a suitable Banach space. Finally, the Cole-Hopf transformation is used to derive an analytical solution in the case when the viscosity is constant. This solution turns out to be very ill-conditioned for numerical evaluations. The solution can be rewritten with the Poisson summation formula. Comparisons to a finite difference solution are done.

From a given velocity field u^*, a flow field that satisfies a given differential equation and minimize some norm is determined. The gradient for the optimization is updated using adjoint technique. The numerical solution of the non-linear partial differential equation is done using a multigrid scheme. The test case shows promising results. The method handles missing data as well as disturbances.

This chapter discusses reconstruction of velocity data, using optimization. There is a growing interest in obtaining velocity data on a higher temporal and/or spatial resolution than is currently possible to measure. The problem originates from a vast array of topics—such as meteorology, hydrology, wind tunnel, or water tunnel experiments—and from noninvasive medical measurement devices, such as 3D time-resolved-phase-contrast magnetic resonance imaging. The rapid development in computer performance gave birth to new methods, based on optimization and simultaneous numerical solution of partial differential equations, well-suited for the task of up-sampling. The data may be of several kinds—low spatial and/or temporal resolution with or without areas of missing and/or uncertain data. It determines a flow field that satisfies a given differential equation and minimize some norm from a given velocity field. The gradient for the optimization can be updated through adjoint technique. The numerical solution of the nonlinear partial differential equation can be done through a multigrid scheme. The method handles missing data as well as disturbances.

For a given field u^*, a field u governed by a nonlinear partial differential equation and minimizing ||u - u^*||₂ is determined. The initial condition is used as control variable and the gradient for the minimization is based on adjoint technique. Continuous and discrete gradient formulations are compared. The method handles missing as well as noisy data.