Development of low dissipative high order filter schemes for multiscale Navier–Stokes/MHD systems

Recent progress in the development of a class of low dissipative high order (fourth-order or higher) filter schemes for multiscale Navier–Stokes, and ideal and non-ideal magnetohydrodynamics (MHD) systems is described. The four main features of this class of schemes are: (a) multiresolution wavelet decomposition of the computed flow data as sensors for adaptive numerical dissipative control, (b) multistep filter to accommodate efficient application of different numerical dissipation models and different spatial high order base schemes, (c) a unique idea in solving the ideal conservative MHD system (a non-strictly hyperbolic conservation law) without having to deal with an incomplete eigensystem set while at the same time ensuring that correct shock speeds and locations are computed, and (d) minimization of the divergence of the magnetic field numerical error. By design, the flow sensors, different choice of high order base schemes and numerical dissipation models are stand-alone modules. A whole class of low dissipative high order schemes can be derived at ease, making the resulting computer software very flexible with widely applicable. Performance of multiscale and multiphysics test cases are illustrated with many levels of grid refinement and comparison with commonly used schemes in the literature

Adaptive filtering and limiting in compact high order methods for multiscale gas dynamics and MHD systems

The adaptive multistep linear and nonlinear filters for multiscale shock/turbulence gas dynamics and magnetohydrodynamics (MHD) flows of the authors are extended to include compact high order central differencing as the spatial base scheme. The adaptive mechanism makes used of multiresolution wavelet decomposition of the computed flow data as sensors for numerical dissipative control. The objective is to expand the work initiated in [Yee HC, Sjo¨green B. Nonlinear filtering in compact high order schemes. In: Proceedings of the 19th ICNSP and 7th APPTC conference; 2005; J Plasma Phys 2006;72:833–36] and compare the performance of adaptive multistep filtering in compact high order schemes with adaptive filtering in standard central (non-compact) schemes for multiscale problems containing shock waves

Resolving Entropy Growth from Iterative Methods

We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton's method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider Burgers' equation and nonlinear dispersive wave equations. We find that entropy conservation results in more accurate numerical solutions than non-conservative schemes, even when the tolerance is an order of magnitude larger.Comment: 25 pages, 6 figure

Analyses and Applications of the Peaceman--Rachford and Douglas--Rachford Splitting Schemes

Splitting methods are widely used as temporal discretizations of evolution equations. Such methods usually constitute competitive choices whenever a vector field can be split into a sum of two or more parts that each generates a flow easier to compute or approximate than the flow of the sum. In the research presented in this Licentiate thesis we consider dissipative evolution equations with vector fields given by unbounded operators. Dynamical systems that fit into this framework can for example be found among Hamiltonian systems and parabolic and hyperbolic partial differential equations (PDEs). The goal of the presented research is to perform convergence analyses for the lternating direction implicit (ADI) methods in the setting of dissipative operators. In this context these methods are known to possess excellent stability properties. Additionally, they generate easily computable numerical flows and are ideal choices for studying convergence to stationary solutions, a property related to their favorable local error structure. In this thesis we consider the Peaceman--Rachford and Douglas--Rachford schemes, which were the first ADI methods to be constructed and to this day are the most representative members of the ADI method class. We perform convergence studies for the Peaceman--Rachford and Douglas--Rachford schemes when applied to semilinear, dissipative evolution equations, that is, when the vector fields are given by the sum of a linear and a nonlinear dissipative operator. Optimal convergence orders are proven when the solution is sufficiently regular. With less regularity present we are still able to prove convergence, however of suboptimal order or without order. In contrast to previous convergence order analyses we do not assume Lipschitz continuity of the nonlinear operator. In the context of linear, dissipative evolution equations we consider full space-time discretizations. We assume that the full discretization is given by combining one of the two aforementioned ADI methods with a general, converging spatial discretization method. In this setting we prove optimal, simultaneous space-time convergence orders. Advection-diffusion-reaction models, encountered in physics, chemistry, and biology are important examples of dissipative evolution equations. In this thesis we present such a model describing the growth of axons in nerve cells. The model consists of a parabolic PDE, which has a non-trivial coupling to nonlinear ordinary differential equations via a moving boundary, which is part of the solution. Since additionally the biological model parameters imply a wide range of scales, both in time and space, the application of a numerical method is involved. We make an argument for a discretization consisting of a splitting which is integrated by the Peaceman--Rachford scheme. The choice is motivate by the results of some numerical experiments