Daniel Appelö

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

A new absorbing layer for elastic waves

2 \times 2

A high-order super-grid-scale absorbing layer and its application to linear hyperbolic systems

systems in (2+1) dimensions.

Computational Modeling and Experiments of Natural Convection for a Titan Montgolfiere

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.

Experiments with Hermite Methods for Simulating Compressible Flows: Runge-Kutta Time-Stepping and Absorbing Layers

We continue the development of the super-grid-scale model initiated in [T. Colonius, H. Ran, A super-grid-scale model for simulating compressible flow on unbounded domains, J. Comput. Phys. 182 (1) (2002) 191-212] and consider its application to linear hyperbolic systems. The super-grid-scale model consists of two parts: reduction of an unbounded to a bounded domain by a smooth coordinate transformation and a damping of those scales. For linear problems the super-grid scales are analogous to spurious numerical waves. We damp these waves by high-order undivided differences. We compute reflection coefficients for different orders of the damping and find that significant improvements are obtained when high-order damping is used. In numerical experiments with Maxwells equations, we show that when the damping is of high order, the error from the boundary condition converges at the order of the interior scheme. We also demonstrate that the new method achieves perfectly matched layer-like accuracy. When applied to linear hyperbolic systems the stability of the super-grid-scale method follows from its construction. This makes our method particularly suitable for problems for which perfectly matched layers are unstable. We present results for two such problems: elastic waves in anisotropic media and isotropic elastic waves in wave guides with traction-free surfaces.

A hybrid Hermite-discontinuous Galerkin method for hyperbolic systems with application to Maxwell's equations

Computational models are developed to predict the natural convection heat transfer and buoyancy for a Montgolfiere under conditions relevant to the Titan atmosphere. Idealized single and double-walled balloon geometries are simulated using algorithms suitable for both laminar and (averaged) turbulent convection. Steady-state performance results are compared to existing heat transfer coefficient correlations. The laminar results, in particular, are used to test the validity of the correlations in the absence of uncertainties associated with turbulence modeling. Some discrepancies are observed, especially for convection in the gap, and appear to be primarily associated with temperature nonuniformity on the balloon surface. The predicted buoyancy for the single-walled balloon in the turbulent convection regime, predicted with a standard k ǫ turbulence model, was within 10% of predictions based on the empirical correlations. There was also good agreement with recently conducted experiments in a cryogenic facility designed to simulate the Titan atmosphere.

A New Discontinuous Galerkin Formulation for Wave Equations in Second-Order Form

In this work we report on various enhancements to our highly accurate compressible flow solver based on Hermite interpolation methods. Our focus here is on two issues: more efficient time-stepping and the implementation of various damping layers. To enhance the efficiency of our time-stepping algorithms, we switch to a Runge-Kutta method from the Taylor series method used in previous works. Our motivation for switching to a low order (4th) method in time is twofold. For many fluid flow problems, though less commonly in aeroacoustic applications, the consensus is that ”the spatial errors will dominate”. Then we may benefit by decreasing the temporal accuracy. Second, even when high temporal accuracy is needed, taking a larger number of Runge-Kutta substeps can still be cheaper than a single Taylor step. We present experiments that show the validity of these statements. Second, in many applications the computational domain must be accurately truncated, due to the small amplitude effects which are central to the simulation. If this is carried out in a region where the equations can be linearized about a uniform or parallel flow, the perfectly matched layer (PML) is an extremely accurate and efficient approach. However, if the truncation is carried out in a region where nonlinear effects are important, the perfect matching property which is central to the accuracy in the linear case no longer strictly holds. Nonetheless, as shown by Hu, PML can still be effective. Here we test implementations of three damping layers, a PML which is a nonlinearization of the layer studied by the authors, the supergrid layer of Colonius and Ran, as well as the simple damping layer proposed for linear problems by Israeli and Orszag and recently analyzed in the weakly nonlinear case by Bodony.

A Fourth-Order Accurate Embedded Boundary Method for the Wave Equation

A high order discretization strategy for solving hyperbolic initial-boundary value problems on hybrid structured-unstructured grids is proposed. The method leverages the capabilities of two distinct families of polynomial elements: discontinuous Galerkin discretizations which can be applied on elements of arbitrary shape, and Hermite discretizations which allow highly efficient implementations on staircased Cartesian grids. We demonstrate through numerical experiments in 1+1 and 2+1 dimensions that the hybridized method is stable and efficient.

Automatic symmetrization and energy estimates using local operators for partial differential equations

We develop and analyze a new strategy for the spatial discontinuous Galerkin discretization of wave equations in second-order form. The method features a direct, mesh-independent approach to defining interelement fluxes. Both energy-conserving and upwind discretizations can be devised. We derive a priori error estimates in the energy norm for certain fluxes and present numerical experiments showing that optimal convergence in

Hermite Methods for Aeroacoustics: Recent Progress

L^2