An Implicit–Explicit Scheme Accurate at Low Mach Numbers for the Wave Equation System

2018 ◽  
pp. 97-109 ◽  
Author(s):  
K. R. Arun ◽  
A. J. Das Gupta ◽  
S. Samantaray
Keyword(s):  
Wave Equation ◽  
Equation System ◽  
Explicit Scheme ◽  
Mach Numbers

Application of unsplit convolutional perfectly matched layer for scalar arbitrarily wide-angle wave equation

Geophysics ◽  
2014 ◽  
Vol 79 (6) ◽  
pp. T313-T321 ◽  
Author(s):  
Hanming Chen ◽  
Hui Zhou ◽  
Yanqi Li
Keyword(s):  
Wave Equation ◽  
Computational Cost ◽  
Grazing Incidence ◽  
Equation System ◽  
Perfectly Matched Layer ◽  
Staggered Grid ◽  
Complex Frequency ◽  
Wide Angle ◽  
First Order ◽  
General Complex

A classical split perfectly matched layer (PML) method has recently been applied to the scalar arbitrarily wide-angle wave equation (AWWE) in terms of displacement. However, the classical split PML obviously increases computational cost and cannot efficiently absorb waves propagating into the absorbing layer at grazing incidence. Our goal was to improve the computational efficiency of AWWE and to enhance the suppression of edge reflections by applying a convolutional PML (CPML). We reformulated the original AWWE as a first-order formulation and incorporated the CPML with a general complex frequency shifted stretching operator into the renewed formulation. A staggered-grid finite-difference (FD) method was adopted to discretize the first-order equation system. For wavefield depth continuation, the first-order AWWE with the CPML saved memory compared with the original second-order AWWE with the conventional split PML. With the help of numerical examples, we verified the correctness of the staggered-grid FD method and concluded that the CPML can efficiently absorb evanescent and propagating waves.

scholarly journals Port-Hamiltonian Representation and Discretization of Undamped Wave Equation System

2016 ◽  
Vol 49 (8) ◽  
pp. 309-314 ◽  
Author(s):  
Qingqing Xu ◽  
Stevan Dubljevic
Keyword(s):  
Wave Equation ◽  
Equation System

scholarly journals On the Stability of Evolution Galerkin Schemes Applied to a Two‐Dimensional Wave Equation System

2006 ◽  
Vol 44 (4) ◽  
pp. 1556-1583 ◽  
Author(s):  
M. Lukáčová‐Medviďová ◽  
G. Warnecke ◽  
Y. Zahaykah
Keyword(s):  
Wave Equation ◽  
Equation System ◽  
Two Dimensional ◽  
The Stability ◽  
Dimensional Wave

Boundary Backstepping Control of Flow-Induced Vibrations of a Membrane at High Mach Numbers

2015 ◽  
Vol 137 (8) ◽  
Author(s):  
Aziz Sezgin ◽  
Miroslav Krstic
Keyword(s):  
Wave Equation ◽  
Original System ◽  
Target System ◽  
Spanwise Direction ◽  
Infinite Length ◽  
Flow Induced Vibration ◽  
Trailing Edge ◽  
Flow Induced Vibrations ◽  
Mach Numbers ◽  
High Mach

We design a controller for flow-induced vibrations of an infinite-band membrane, with flow running across the band and only above it, and with actuation only on the trailing edge of the membrane. Due to the infinite length of the membrane, the dynamics of the membrane in the spanwise direction are neglected, namely, we employ a one-dimensional (1D) model that focuses on streamwise vibrations. This framework is inspired by a flow along an airplane wing with actuation on the trailing edge. The model of the flow-induced vibration is given by a wave partial differential equation (PDE) with an antidamping term throughout the 1D domain. Such a model is based on linear aeroelastic theory for Mach numbers above 0.8. To design a controller, we introduce a three-stage backstepping transformation. The first stage gets the system to a critically antidamped wave equation, changing the stiffness coefficient's value but not its sign. The second stage changes the system from a critically antidamped to a critically damped equation with an arbitrary damping coefficient. The third stage adjusts stiffness arbitrarily. The controller and backstepping transformation map the original system into a target system given by a wave equation with arbitrary positive damping and stiffness.

Multipoint Explicit Differencing (MED) for Time Integrations of the Wave Equation

2007 ◽  
Vol 135 (11) ◽  
pp. 3862-3875 ◽  
Author(s):  
Celal S. Konor ◽  
Akio Arakawa
Keyword(s):  
Wave Equation ◽  
Finite Number ◽  
Multilevel Models ◽  
Horizontal Resolution ◽  
Explicit Scheme ◽  
Implicit Scheme ◽  
Courant Number ◽  
Time Step ◽  
Wide Range ◽  
Grid Points

Abstract For time integrations of the wave equation, it is desirable to use a scheme that is stable over a wide range of the Courant number. Implicit schemes are examples of such schemes, but they do that job at the expense of global calculation, which becomes an increasingly serious burden as the horizontal resolution becomes higher while covering a large horizontal domain. If what an implicit scheme does from the point of view of explicit differencing is looked at, it is a multipoint scheme that requires information at all grid points in space. Physically this is an overly demanding requirement because wave propagation in the real atmosphere has a finite speed. The purpose of this study is to seek the feasibility of constructing an explicit scheme that does essentially the same job as an implicit scheme with a finite number of grid points in space. In this paper, a space-centered trapezoidal implicit scheme is used as the target scheme as an example. It is shown that an explicit space-centered scheme with forward time differencing using an infinite number of grid points in space can be made equivalent to the trapezoidal implicit scheme. To avoid global calculation, a truncated version of the scheme is then introduced that only uses a finite number of grid points while maintaining stability. This approach of constructing a stable explicit scheme is called multipoint explicit differencing (MED). It is shown that the coefficients in an MED scheme can be numerically determined by single-time-step integrations of the target scheme. With this procedure, it is rather straightforward to construct an MED scheme for an arbitrarily shaped grid and/or boundaries. In an MED scheme, the number of grid points necessary to maintain stability and, therefore, the CPU time needed for each time step increase as the Courant number increases. Because of this overhead, the MED scheme with a large time step can be more efficient than a usual explicit scheme with a smaller time step only for complex multilevel models with detailed physics. The efficiency of an MED scheme also depends on how the advantage of parallel computing is taken.

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