Modeling the scalar wave equation with Nyström methods

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. T151-T158 ◽  
Author(s):  
Jing-Bo Chen
Keyword(s):  
Wave Equation ◽  
Fourth Order ◽  
Second Order ◽  
Scalar Wave ◽  
Nyström Method ◽  
Scalar Wave Equation ◽  
Nystrom Method ◽  
Nyström Methods ◽  
Long Time ◽  
Order Scheme

High-accuracy numerical schemes for modeling of the scalar wave equation based on Nyström methods are developed in this paper. Space is discretized by using the pseudospectral algorithm. For the time discretization, Nyström methods are used. A fourth-order symplectic Nyström method with pseudospectral spatial discretization is presented. This scheme is compared with a commonly used second-order scheme and a fourth-order nonsymplectic Nyström method. For a typical time-step size, the second-order scheme exhibits spatial dispersion errors for long-time simulations, while both fourth-order schemes do not suffer from these errors. Numerical comparisons show that the fourth-order symplectic algorithm is more accurate than the fourth-order nonsymplectic one. The capability of the symplectic Nyström method in approximately preserving the discrete energy for long-time simulations is also demonstrated.

A stability formula for Lax-Wendroff methods with fourth-order in time and general-order in space for the scalar wave equation

Geophysics ◽  
2011 ◽  
Vol 76 (2) ◽  
pp. T37-T42 ◽  
Author(s):  
Jing-Bo Chen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Fourth Order ◽  
Second Order ◽  
High Order ◽  
Scalar Wave ◽  
Scalar Wave Equation ◽  
General Order ◽  
Second Order In Time ◽  
Spatial Derivatives

Based on the formula for stability of finite-difference methods with second-order in time and general-order in space for the scalar wave equation, I obtain a stability formula for Lax-Wendroff methods with fourth-order in time and general-order in space. Unlike the formula for methods with second-order in time, this formula depends on two parameters: one parameter is related to the weights for approximations of second spatial derivatives; the other parameter is related to the weights for approximations of fourth spatial derivatives. When discretizing the mixed derivatives properly, the formula can be generalized to the case where the spacings in different directions are different. This formula can be useful in high-accuracy seismic modeling using the scalar wave equation on rectangular grids, which involves both high-order spatial discretizations and high-order temporal approximations. I also prove the instability of methods obtained by applying high-order finite-difference approximations directly to the second temporal derivative, and this result solves the “Bording’s conjecture.”

scholarly journals THE EASTWOOD–SINGER GAUGE IN EINSTEIN SPACES

2009 ◽  
Vol 06 (04) ◽  
pp. 583-593 ◽  
Author(s):  
GIAMPIERO ESPOSITO ◽  
RAJU ROYCHOWDHURY
Keyword(s):  
Wave Equation ◽  
Fourth Order ◽  
Space Time ◽  
Order Equation ◽  
Scalar Wave ◽  
Scalar Wave Equation ◽  
Fourth Order Equation ◽  
Vector Wave ◽  
Vector Wave Equation ◽  
Einstein Spaces

Electrodynamics in curved space-time can be studied in the Eastwood–Singer gauge, which has the advantage of respecting the invariance under conformal rescalings of the Maxwell equations. Such a construction is here studied in Einstein spaces, for which the Ricci tensor is proportional to the metric. The classical field equations for the potential are then equivalent to first solving a scalar wave equation with cosmological constant, and then solving a vector wave equation where the inhomogeneous term is obtained from the gradient of the solution of the scalar wave equation. The Eastwood–Singer condition leads to a field equation on the potential which is preserved under gauge transformations provided that the scalar function therein obeys a fourth-order equation where the highest-order term is the wave operator composed with itself. The second-order scalar equation is here solved in de Sitter space-time, and also the fourth-order equation in a particular case, and these solutions are found to admit an exponential decay at large time provided that square-integrability for positive time is required. Last, the vector wave equation in the Eastwood–Singer gauge is solved explicitly when the potential is taken to depend only on the time variable.

Seismic scalar wave equation with variable coefficients modeling by a new convolutional differentiator

2010 ◽  
Vol 181 (11) ◽  
pp. 1850-1858 ◽  
Author(s):  
Xiaofan Li ◽  
Tong Zhu ◽  
Meigen Zhang ◽  
Guihua Long
Keyword(s):  
Wave Equation ◽  
Variable Coefficients ◽  
Scalar Wave ◽  
Scalar Wave Equation

scholarly journals MOTION OF MASSIVE AND MASSLESS TEST PARTICLES IN DYADOSPHERE GEOMETRY

2009 ◽  
Vol 24 (16) ◽  
pp. 1277-1287 ◽  
Author(s):  
B. RAYCHAUDHURI ◽  
F. RAHAMAN ◽  
M. KALAM ◽  
A. GHOSH
Keyword(s):  
Wave Equation ◽  
Test Particle ◽  
Massless Particle ◽  
Jacobi Method ◽  
Scalar Wave ◽  
Scalar Wave Equation ◽  
Test Particles

Motion of massive and massless test particle in equilibrium and nonequilibrium case is discussed in a dyadosphere geometry through Hamilton–Jacobi method. Scalar wave equation for massless particle is analyzed to show the absence of superradiance in the case of dyadosphere geometry.

Modal methods for the scalar wave equation

1983 ◽  
pp. 640-655 ◽  
Author(s):  
Allan W. Snyder ◽  
John D. Love
Keyword(s):  
Wave Equation ◽  
Scalar Wave ◽  
Scalar Wave Equation
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