Gunilla Kreiss

PERFECTLY MATCHED LAYERS FOR HYPERBOLIC SYSTEMS: GENERAL FORMULATION, WELL-POSEDNESS AND STABILITY

Since its introduction the perfectly matched layer (PML) has proven to be an accurate and robust method for domain truncation in computational electromagnetics. However, the mathematical analysis of PMLs has been limited to special cases. In particular, the basic question of whether or not a stable PML exists for arbitrary wave propagation problems remains unanswered. In this work we develop general tools for constructing PMLs for first order hyperbolic systems. We present a model with many parameters, which is applicable to all hyperbolic systems and which we prove is well‐posed and perfectly matched. We also introduce an automatic method for analyzing the stability of the model and establishing energy inequalities. We illustrate our techniques with applications to Maxwell’s equations, the linearized Euler equations, and arbitrary

Bounds for threshold amplitudes in subcritical shear flows

2 \times 2

A new absorbing layer for elastic waves

systems in (2+1) dimensions.

Stability of systems of viscous conservation laws

A general theory which can be used to derive bounds on solutions to the Navier-Stokes equations is presented. The behaviour of the resolvent of the linear operator in the unstable half-plane is used to bound the energy growth of the full nonlinear problem. Plane Couette flow is used as an example. The norm of the resolvent in plane Couette flow in the unstable half-plane is proportional to the square of the Reynolds number ( R ). This is now used to predict the asymptotic behaviour of the threshold amplitude below which all disturbances eventually decay. A lower bound is found to be R −21/4 . Examples, obained through direct numerical simulation, give an upper bound on the threshold curve, and predict a threshold of R −1 . The discrepancy is discussed in the light of a model problem.

A conservative level set method for contact line dynamics

A new perfectly matched layer (PML) for the simulation of elastic waves in anisotropic media on an unbounded domain is constructed. Theoretical and numerical results, showing that the stability properties of the present layer are better than previously suggested layers, are presented. In addition, the layer can be formulated with fewer auxiliary variables than the split-field PML.

On the convergence to steady state of solutions of nonlinear hyperbolic-parabolic systems

We consider systems of conservation laws and give conditions for nonlinear stability of viscous shock profiles. The analysis applies to classical shocks of arbitrary strength.

Stability at Nonconforming Grid Interfaces for a High Order Discretization of the Schrödinger Equation

A new model for simulating contact line dynamics is proposed. We apply the idea of driving contact-line movement by enforcing the equilibrium contact angle at the boundary, to the conservative level set method for incompressible two-phase flow [E. Olsson, G. Kreiss, A conservative level set method for two phase flow, J. Comput. Phys. 210 (2005) 225-246]. A modified reinitialization procedure provides a diffusive mechanism for contact-line movement, and results in a smooth transition of the interface near the contact line without explicit reconstruction of the interface. We are able to capture contact-line movement without loosing the conservation. Numerical simulations of capillary dominated flows in two space dimensions demonstrate that the model is able to capture contact line dynamics qualitatively correct.

A Remark on Numerical Errors Downstream of Slightly Viscous Shocks

The Cauchy problem for time-dependent quasi-linear partial differential equations is considered and the nonlinear stability of steady state solutions is investigated.

High Order Stable Finite Difference Methods for the Schrödinger Equation

In this paper we extend the results from our earlier work on stable boundary closures for the Schrödinger equation using the summation-by-parts-simultaneous approximation term (SBP–SAT) method to include stability and accuracy at nonconforming grid interfaces. Stability at the grid interface is shown by the energy method, and the estimates are generalized to multiple dimensions. The accuracy of the grid interface coupling is investigated using normal mode analysis for operators of 2nd and 4th order formal interior accuracy. We show that full accuracy is retained for the 2nd and 4th order operators. The accuracy results are extended to 6th and 8th order operators by numerical simulations, in which case two orders of accuracy is gained with respect to the lower order approximation close to the interface.

Bounds for the Threshold Amplitude for Plane Couette Flow

Lower-order errors downstream of a shock layer have been detected in computations with nonconstant solutions when using higher-order shock capturing schemes in one and two dimensions [B. Engquist and B. Sjogreen, {SIAM J. Numer. Anal., 35 (1998), pp. 2464--2485]. By analyzing the steady-state solution of slightly viscous hyperbolic systems of conservation laws we find that the solution can have an