High-resolution Wave Propagation Research Articles

Numerical discretisation of hyperbolic systems of moment equations describing sedimentation in suspensions of rod-like particles

We present a numerical discretisation of the coupled moment systems, previously introduced in Dahm and Helzel [1], which approximate the kinetic-fluid model by Helzel and Tzavaras [2] for sedimentation in suspensions of rod-like particles for a two-dimensional flow problem and a shear flow problem. We use a splitting ansatz which, during each time step, separately computes the update of the macroscopic flow equation and of the moment system. The proof of the hyperbolicity of the moment systems in [1] suggests solving the moment systems with standard numerical methods for hyperbolic problems, like LeVeque's Wave Propagation Algorithm [3]. The number of moment equations used in the hyperbolic moment system can be adapted to locally varying flow features. An error analysis is proposed, which compares the approximation with 2N+1 moment equations to an approximation with 2N+3 moment equations. This analysis suggests an error indicator which can be computed from the numerical approximation of the moment system with 2N+1 moment equations. In order to use moment approximations with a different number of moment equations in different parts of the computational domain, we consider an interface coupling of moment systems with different resolution. Finally, we derive a conservative high-resolution Wave Propagation Algorithm for solving moment systems with different numbers of moment equations.

A High-Resolution Finite Volume Seismic Model to Generate Seafloor Deformation for Tsunami Modeling

A high-resolution finite volume method approach to incorporating time-dependent slip across rectangular subfaults when modeling general fault geometry is presented. The fault slip is induced by a modification of the Riemann problem to the linear elasticity equations across cell interfaces aligned with the subfaults. This is illustrated in the context of the high-resolution wave-propagation algorithms that are implemented in the open source Clawpack software (www.clawpack.org), but this approach could be easily incorporated into other Riemann solver based numerical methods. Surface deformation results are obtained in both two and three dimensions and compared to those given by the steady-state, homogeneous half-space Okada solution.

Coastal defence through wave farms

The possibility of using wave farms for coastal defence warrants investigation because wave energy is poised to become a major renewable in many countries over the next decades. The fundamental question in this regard is whether a wave farm can be used to reduce beach erosion under storm conditions. If the answer to this question is positive, then a wave farm can have coastal defence as a subsidiary function, in addition to its primary role of producing carbon-free energy. The objective of this work is to address this question by comparing the response of a beach in the face of a storm in two scenarios: with and without the wave farm. For this comparison a set of ad hoc impact indicators is developed: the bed level impact (BLI), beach face eroded area (FEA), non-dimensional erosion reduction (NER), and mean cumulative eroded area (CEA); and their values are determined by means of two coupled models: a high-resolution wave propagation model (SWAN) and a coastal processes model (XBeach). The study is conducted through a case study: Perranporth Beach (UK). Backed by a well-developed dune system, Perranporth has a bar between −5m and −10m. The results show that the wave farm reduces the eroded volume by as much as 50% and thus contributes effectively to coastal protection. This synergy between marine renewable energy and coastal defence may well contribute to improving the viability of wave farms through savings in conventional coastal protection.

A mixture-energy-consistent six-equation two-phase numerical model for fluids with interfaces, cavitation and evaporation waves

We model liquid–gas flows with cavitation by a variant of the six-equation single-velocity two-phase model with stiff mechanical relaxation of Saurel–Petitpas–Berry (Saurel et al., 2009) [9]. In our approach we employ phasic total energy equations instead of the phasic internal energy equations of the classical six-equation system. This alternative formulation allows us to easily design a simple numerical method that ensures consistency with mixture total energy conservation at the discrete level and agreement of the relaxed pressure at equilibrium with the correct mixture equation of state. Temperature and Gibbs free energy exchange terms are included in the equations as relaxation terms to model heat and mass transfer and hence liquid–vapor transition. The algorithm uses a high-resolution wave propagation method for the numerical approximation of the homogeneous hyperbolic portion of the model. In two dimensions a fully-discretized scheme based on a hybrid HLLC/Roe Riemann solver is employed. Thermo-chemical terms are handled numerically via a stiff relaxation solver that forces thermodynamic equilibrium at liquid–vapor interfaces under metastable conditions. We present numerical results of sample tests in one and two space dimensions that show the ability of the proposed model to describe cavitation mechanisms and evaporation wave dynamics.

High resolution wave propagation algorithm for one dimensional hyperbolic system

High resolution wave propagation algorithm for one dimensional hyperbolic system

High resolution tsunami modelling for the evaluation of potential risk areas in Setúbal (Portugal)

Abstract. The use of high resolution hydrodynamic modelling to simulate the potential effects of tsunami events can provide relevant information about the most probable inundation areas. Moreover, the consideration of complementary data such as the type of buildings, location of priority equipment, type of roads, enables mapping of the most vulnerable zones, computing of the expected damage on man-made structures, constrain of the definition of rescue areas and escape routes, adaptation of emergency plans and proper evaluation of the vulnerability associated with different areas and/or equipment. Such an approach was used to evaluate the specific risks associated with a potential occurrence of a tsunami event in the region of Setúbal (Portugal), which was one of the areas most seriously affected by the 1755 tsunami. In order to perform an evaluation of the hazard associated with the occurrence of a similar event, high resolution wave propagation simulations were performed considering different potential earthquake sources with different magnitudes. Based on these simulations, detailed inundation maps associated with the different events were produced. These results were combined with the available information on the vulnerability of the local infrastructures (building types, roads and streets characteristics, priority buildings) in order to impose restrictions in the production of high-scale potential damage maps, escape routes and emergency routes maps.

An unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations

Numerical methods for solving the ideal magnetohydrodynamic (MHD) equations in more than one space dimension must either confront the challenge of controlling errors in the discrete divergence of the magnetic field, or else be faced with nonlinear numerical instabilities. One approach for controlling the discrete divergence is through a so-called constrained transport method, which is based on first predicting a magnetic field through a standard finite volume solver, and then correcting this field through the appropriate use of a magnetic vector potential. In this work we develop a constrained transport method for the 3D ideal MHD equations that is based on a high-resolution wave propagation scheme. Our proposed scheme is the 3D extension of the 2D scheme developed by Rossmanith [J.A. Rossmanith, An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows, SIAM J. Sci. Comput. 28 (2006) 1766], and is based on the high-resolution wave propagation method of Langseth and LeVeque [J.O. Langseth, R.J. LeVeque, A wave propagation method for threedimensional hyperbolic conservation laws, J. Comput. Phys. 165 (2000) 126]. In particular, in our extension we take great care to maintain the three most important properties of the 2D scheme: (1) all quantities, including all components of the magnetic field and magnetic potential, are treated as cell-centered; (2) we develop a high-resolution wave propagation scheme for evolving the magnetic potential; and (3) we develop a wave limiting approach that is applied during the vector potential evolution, which controls unphysical oscillations in the magnetic field. One of the key numerical difficulties that is novel to 3D is that the transport equation that must be solved for the magnetic vector potential is only weakly hyperbolic. In presenting our numerical algorithm we describe how to numerically handle this problem of weak hyperbolicity, as well as how to choose an appropriate gauge condition. The resulting scheme is applied to several numerical test cases.

Nature of polarization in wide-bandgap semiconductor detectors under high-flux irradiation: Application to semi-insulatingCd1−xZnxTe

In this paper, we theoretically investigate the mechanism of polarization in wide-bandgap semiconductor radiation detectors under high-flux x-ray irradiation. Our general mathematical model of the defect structure within the bandgap is a system of balance laws based on carrier transport and defect transition rates, coupled together with the Poisson equation for the electric potential. The dynamical system is self-consistently evolved in time using a high-resolution wave propagation numerical algorithm. Through simulation, we identify and present a sequence of dynamics that determines a critical flux of photons above which polarization effects dominate. Using the experience gained through numerical simulation of the full set of equations, we derive a reduced system of conservation laws that describe the dominant dynamics. A multiple scale perturbation analysis of the reduced system is shown to yield an analytical dependence of the maximum sustainable flux on key material, detector, and operating parameters. The predicted dependencies are validated for $16\ifmmode\times\else\texttimes\fi{}16\phantom{\rule{0.3em}{0ex}}\text{pixel}$ CdZnTe monolithic detector arrays subjected to a high-flux $120\phantom{\rule{0.3em}{0ex}}\mathrm{kVp}$ x-ray source.

Logically Rectangular Grids and Finite Volume Methods for PDEs in Circular and Spherical Domains

We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere, and the three-dimensional ball. The grids are logically rectangular and the computational domain is a single Cartesian grid. Compared to alternative approaches based on a multiblock data structure or unstructured triangulations, this approach simplifies the implementation of numerical methods and the use of adaptive refinement. A more general domain with a smooth boundary can be gridded by composing one of the mappings from this paper with another smooth mapping from the circle or sphere to the desired domain. Although these grids are highly nonorthogonal, we show that the high-resolution wave-propagation algorithm implemented in clawpack can be used effectively to approximate hyperbolic problems on these grids. Since the ratio between the largest and smallest grids is below 2 for most of our grid mappings, explicit finite volume methods such as the wave-propagation algorithm do not suffer from the center or pole singularities that arise with polar or latitude-longitude grids. Numerical test calculations illustrate the potential use of these grids for a variety of applications including Euler equations, shallow water equations, and acoustics in a heterogeneous medium. Pattern formation from a reaction-diffusion equation on the sphere is also considered. All examples are implemented in the clawpack software package and full source code is available on the web, along with MATLAB routines for the various mappings.

Two-fluid physics and field-reversed configurations

In this paper, algorithms for the solution of two-fluid plasma equations are presented and applied to the study of field-reversed configurations (FRCs). The two-fluid model is more general than the often used magnetohydrodynamic (MHD) model. The model takes into account electron inertia, charge separation, and the full electromagnetic field equations, and it allows for separate electron and ion motion. The algorithm presented is the high-resolution wave propagation scheme. The wave propagation method is based on solutions to the Riemann problem at cell interfaces. Operator splitting is used to incorporate the Lorentz and electromagnetic source terms. The algorithms are benchmarked against the Geospace Environmental Modeling Reconnection Challenge problem. Equilibrium of FRC is studied. It is shown that starting from a MHD equilibrium produces a relaxed two-fluid equilibrium with strong flows at the FRC edges due to diamagnetic drift. The azimuthal electron flow causes lower-hybrid drift instabilities (LHDI), which can be captured if the ion gyroradius is well resolved. The LHDI is known to be a possible source of anomalous resistivity in many plasma configurations. LHDI simulations are performed in slab geometries and are compared to recent experimental results.

A volume-fraction based algorithm for hybrid barotropic and non-barotropic two-fluid flow problems

The aim of this paper is to describe a simple Eulerian interface-capturing approach for the efficient numerical resolution of a hybrid barotropic and non-barotropic two-fluid flow problem in more than one space dimension. We use the compressible Euler equations as a model system with the thermodynamic property of each of the barotropic and non-barotropic fluid components characterized by the Tait and Noble–Abel equations of state, respectively. The algorithm is based on a volume fraction formulation of the equations together with an extended equation of state that is devised to give an approximate treatment for the mixture of more than one fluid component within a grid cell. A standard high-resolution wave propagation method is employed to solve the proposed two-fluid model with the dimensional-splitting technique incorporated in the method for multidimensional problems. Several numerical results are presented in one and two space dimensions that show the feasibility of the algorithm as applied to a reasonable class of practical problems without the occurrence of any spurious oscillation in the pressure near the smeared material interfaces. This includes, in particular, solutions for a study on the variation of the jet velocity with the incident shock pressure arising from the collapse of an air cavity in water under a shock wave.

A high resolution wave propagation scheme for ideal Two-Fluid plasma equations

Algorithms for the solution of the five-moment ideal Two-Fluid equations are presented. The ideal Two-Fluid model is more general than the often used magnetohydrodynamic (MHD) model. The model takes into account electron inertia effects, charge separation and the full electromagnetic field equations and allows for separate electron and ion motion. The algorithm presented is the high resolution wave propagation method. The wave propagation method is based on solutions to the Riemann problem at cell interfaces. Operator splitting is used to incorporate the Lorentz and electromagnetic source terms. To preserve the divergence constraints on the electric and magnetic fields two different approaches are used. In the first approach Maxwell equations are rewritten in their mixed-potential form. In the second approach the so-called perfectly hyperbolic form of Maxwell equations are used which explicitly incorporate the divergence equations into the time stepping scheme. The algorithm is applied to a one-dimensional Riemann problem, ion-acoustic soliton propagation and magnetic reconnection. In each case Two-Fluid physics described by the ideal Two-Fluid model is highlighted.

An Unstaggered, High‐Resolution Constrained Transport Method for Magnetohydrodynamic Flows

The ideal magnetohydrodynamic (MHD) equations are important in modeling phenomena in a wide range of applications, including space weather, solar physics, laboratory plasmas, and astrophysical fluid flows. Numerical methods for the MHD equations must confront the challenge of producing approximate solutions that remain accurate near shock waves and that satisfy a divergence-free constraint on the magnetic field. Failure to accomplish this often leads to unphysical solutions. In this paper, a high-resolution wave propagation method is developed that utilizes a novel constrained transport technique to keep the magnetic field divergence-free. This approach is based on directly solving the magnetic potential equation in conjunction with a new limiting strategy to obtain a nonoscillatory magnetic field. It is demonstrated in this work that an unstaggered definition of the divergence is the correct one to use in the case of wave propagation methods. Therefore, we solve the magnetic potential equation on the same grid as the MHD equations; hence the usual grid staggering that is found in constrained transport methods is eliminated. We demonstrate through truncation error analysis and direct numerical simulation that the resulting method is second order accurate in space and time for smooth solutions and nonoscillatory near shocks and other discontinuities. The resulting numerical method has been implemented as an extension to the clawpack software package and can be freely downloaded from the Web.

High‐Resolution Finite Volume Methods for Dusty Gas Jets and Plumes

We consider a model for dusty gas flow that consists of the compressible Euler equations for the gas coupled to a similar (but pressureless) system of equations for the mass, momentum, and energy of the dust. These sets of equations are coupled via drag terms and heat transfer. A high-resolution wave-propagation algorithm is used to solve the equations numerically. The one-dimensional algorithm is shown to give agreement with a shock tube test problem in the literature. The two-dimensional algorithm has been applied to model explosive volcanic eruptions in which an axisymmetric jet of hot dusty gas is injected into the atmosphere and the expected behavior is observed at two different vent velocities. The methodology described here, with extensions to three dimensions and adaptive mesh refinement, is being used for more detailed studies of volcanic jet processes.

A wave-propagation based volume tracking method for compressible multicomponent flow in two space dimensions

We present a simple volume-of-fluid approach to interface tracking for inviscid compressible multicomponent flow problems in two space dimensions. The algorithm uses a uniform Cartesian grid with some grid cells subdivided by tracked interfaces, approximately aligned with the material interfaces in the flow field. A standard volume-moving procedure that consists of two basic steps: (1) the update of a discrete set of volume fractions from the current time to the next and (2) the reconstruction of interfaces from the resulting volume fractions, is employed to find the new location of the tracked interfaces in piecewise linear form at the end of a time step. As in the previous work by LeVeque and the author to front tracking based on a surface-moving procedure (R.J. LeVeque, K.-M. Shyue, Two-dimensional front tracking based on high-resolution wave propagation methods, J. Comput. Phys. 123 (1996) 354–368), a conservative high-resolution wave propagation method is applied on the resulting slightly non-uniform grid to update all the cell values, while the stability of the method is maintained by using a large time step idea even in the presence of small cells and the use of a time step with respect to the uniform grid cells. We validate our algorithm by performing the simulation of a Mach 1.22 shock wave in air over a circular R22 gas bubble, where sensible agreement of some key flow features of the computed solutions are observed when direct comparison of our results are made with the existing experimental and numerical ones that appear in the literature. Other computations are also presented that show the feasibility of the algorithm together with a mixture type of the model equations developed by the author (K.-M. Shyue, A fluid-mixture type algorithm for compressible multicomponent flow with Mie–Grüneisen equation of state, J. Comput. Phys. 171 (2001) 678–707) for practical multicomponent problems with general compressible materials characterized by a Mie–Grüneisen form of the equation of state.

H-Box Methods for the Approximation of Hyperbolic Conservation Laws on Irregular Grids

We study generalizations of the high-resolution wave propagation algorithm for the approximation of hyperbolic conservation laws on irregular grids that have a time step restriction based on a reference grid cell length that can be orders of magnitude larger than the smallest grid cell arising in the discretization. This Godunov-type scheme calculates fluxes at cell interfaces by solving Riemann problems defined over boxes of a reference grid cell length h. We discuss stability and accuracy of the resulting so-called h-box methods for one-dimensional systems of conservation laws. An extension of the method for the two-dimensional case, which is based on the multidimensional wave propagation algorithm, is also described.

A Wave Propagation Method for Conservation Laws and Balance Laws with Spatially Varying Flux Functions

We study a general approach to solving conservation laws of the form qt+f(q,x)x=0, where the flux function f(q,x) has explicit spatial variation. Finite-volume methods are used in which the flux is discretized spatially, giving a function fi(q) over the ith grid cell and leading to a generalized Riemann problem between neighboring grid cells. A high-resolution wave-propagation algorithm is defined in which waves are based directly on a decomposition of flux differences fi(Qi-1(Qi-1 ) into eigenvectors of an approximate Jacobian matrix. This method is shown to be second-order accurate for smooth problems and allows the application of wave limiters to obtain sharp results on discontinuities. Balance laws $q_t+f(q,x)_x=\psi(q,x)$ are also considered, in which case the source term is used to modify the flux difference before performing the wave decomposition, and an additional term is derived that must also be included to obtain full accuracy. This method is particularly useful for quasi-steady problems close to steady state.

A Fluid-Mixture Type Algorithm for Compressible Multicomponent Flow with Mie–Grüneisen Equation of State

A simple interface-capturing approach proposed previously by the author for efficient numerical resolution of multicomponent problems with a van der Waals fluid [J. Comput. Phys., 156 (1999), pp. 43–88] is extended to a more general case with real materials characterized by a Mie–Grüneisen equation of state. As before, the flow regime of interests is assumed to be homogeneous with no jumps in the pressure and velocity (the normal component of it) across the interfaces that separate two regions of different fluid components. The algorithm uses a mixture type of the model system that is formed by combining the Euler equations of gas dynamics for the basic conserved variables and an additional set of effective equations for the problem-dependent material quantities. In this approach, the latter equations are introduced in the algorithm primarily for an easy computation of the pressure from the equation of state, and are derived so as to ensure a consistent modeling of the energy equation near the interfaces where two or more fluid components are present in a grid cell, and also the fulfillment of the mass equation in the other single component regions. A standard high-resolution wave propagation method designed originally for single component flows is generalized to solve the proposed system for multicomponent flows, giving an efficient implementation of the algorithm. Several numerical results are presented in both one and two space dimensions that show the feasibility of the method with the Roe Riemann solver as applied to a reasonable class of practical problems without introducing any spurious oscillations in the pressure near the interfaces. This includes results obtained using a multicomponent version of the AMRCLAW software package of Berger and LeVeque for the simulation of the impact of an underwater aluminum plate to a copper plate in two space dimensions.

The MoT-ICE: A New High-Resolution Wave-Propagation Algorithm for Multidimensional Systems of Conservation Laws Based on Fey's Method of Transport

Fey's Method of Transport (MoT) is a multidimensional flux-vector-splitting scheme for systems of conservation laws. Similarly to its one-dimensional forerunner, the Steger–Warming scheme, and several other upwind finite-difference schemes, the MoT suffers from an inconsistency at sonic points when used with piecewise-constant reconstructions. This inconsistency is due to a cell-centered evolution scheme, which we call MoT-CCE, that is used to propagate the waves resulting from the flux-vector-splitting step. Here we derive new first-order- and second-order-consistent characteristic schemes based on interface-centered evolution, which we call MoT-ICE. We prove consistency at all points, including the sonic points. Moreover, we simplify Fey's wave decomposition by distinguishing clearly between a linearization and a decomposition step. Numerical experiments confirm the stability and accuracy of the new schemes. Owing to the simplicity of the two new ingredients of the MoT-ICE, its second-order version is several times faster than that of the MoT-CCE.

Adaptive Mesh Refinement Using Wave-Propagation Algorithms for Hyperbolic Systems

An adaptive mesh refinement algorithm developed for the Euler equations of gas dynamics has been extended to employ high-resolution wave-propagation algorithms in a more general framework. This allows its use on a variety of new problems, including hyperbolic equations not in conservation form, problems with source terms or capacity functions, and logically rectangular curvilinear grids. This framework requires a modified approach to maintaining consistency and conservation at grid interfaces, which is described in detail. The algorithm is implemented in the AMRCLAW package, which is freely available.