High order modified differential equation of the Beam–Warming method, II. The dissipative features

2020 ◽  
Vol 35 (3) ◽  
pp. 175-185
Author(s):  
Yurii Shokin ◽  
Ireneusz Winnicki ◽  
Janusz Jasinski ◽  
Slawomir Pietrek
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wind Speed ◽  
Difference Scheme ◽  
Advection Equation ◽  
Explicit Difference Scheme ◽  
Eighth Order ◽  
Differential Approximation ◽  
Partial Differential ◽  
Linear Advection Equation

AbstractThis paper is a continuation of [38]. The analysis of the modified partial differential equation (MDE) of the constant-wind-speed linear advection equation explicit difference scheme up to the eighth-order derivatives is presented. In this paper the authors focus on the dissipative features of the Beam–Warming scheme. The modified partial differential equation is presented in the so-called Π-form of the first differential approximation. The most important part of this form includes the coefficients μ (p) at the space derivatives. Analysis of these coefficients provides indications of the nature of the dissipative errors. A fragment of the stencil for determining the modified differential equation for the Beam–Warming scheme is included. The derived and presented coefficients μ (p) as well as the analysis of the dissipative features of this scheme on the basis of these coefficients have not been published so far.

High order modified differential equation of the Beam–Warming method, I. The dispersive features

2020 ◽  
Vol 35 (2) ◽  
pp. 83-94 ◽  
Author(s):  
Yurii Shokin ◽  
Ireneusz Winnicki ◽  
Janusz Jasinski ◽  
Slawomir Pietrek
Keyword(s):  
Differential Equation ◽  
Order Equation ◽  
Advection Equation ◽  
Explicit Difference Scheme ◽  
Eighth Order ◽  
Differential Approximation ◽  
Phase And Group Velocities ◽  
Derivatives Of ◽  
Modified Equation ◽  
Group Velocities

Abstract The analysis of the modified partial differential equation (MDE) of the constant wind speed advection equation explicit difference scheme up to the eighth order with respect to both space and time derivatives is presented. So far, in majority of publications this modified equation has been derived mainly as a fourth-order equation. The MDE is presented in the so-called Π-form of the first differential approximation. This form includes only the space derivatives of higher order p and their coefficients μ(p). Analysis of these coefficients provides indications of the nature of the dissipative and dispersive errors. A fragment of the stencil for determining the modified differential equation up to the eighth-order MDE for the second-order Beam–Warming scheme is included. The derived coefficients μ(p) as well as the analysis of the phase shift errors, the phase and group velocities and dispersive features on the basis of these coefficients have not been published so far. The dissipative features of this method we present in [33].

Eulerian partial-differential-equation methods for complex-valued eikonals in attenuating media

Geophysics ◽  
2021 ◽  
pp. 1-53
Author(s):  
Jiangtao Hu ◽  
Jianliang Qian ◽  
Jian Song ◽  
Min Ouyang ◽  
Junxing Cao ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Ray Tracing ◽  
Seismic Waves ◽  
Real Space ◽  
Advection Equation ◽  
Partial Differential ◽  
The Real ◽  
Eikonal Function ◽  
Complex Valued

Seismic waves in earth media usually undergo attenuation, causing energy losses and phase distortions. In the regime of high-frequency asymptotics, a complex-valued eikonal is an essential ingredient for describing wave propagation in attenuating media, where the real and imaginary parts of the eikonal function capture dispersion effects and amplitude attenuation of seismic waves, respectively. Conventionally, such a complex-valued eikonal is mainly computed either by tracing rays exactly in complex space or by tracing rays approximately in real space so that the resulting eikonal is distributed irregularly in real space. However, seismic data processing methods, such as prestack depth migration and tomography, usually require uniformly distributed complex-valued eikonals. Therefore, we propose a unified framework to Eulerianize several popular approximate real-space ray-tracing methods for complex-valued eikonals so that the real and imaginary parts of the eikonal function satisfy the classical real-space eikonal equation and a novel real-space advection equation, respectively, and we dub the resulting method the Eulerian partial-differential-equation method. We further develop highly efficient high-order methods to solve these two equations by using the factorization idea and the Lax-Friedrichs weighted essentially non-oscillatory (WENO) schemes. Numerical examples demonstrate that the proposed method yields highly accurate complex-valued eikonals, analogous to those from ray-tracing methods. The proposed methods can be useful for migration and tomography in attenuating media.

INTENSITY-BASED MODELS FOR PRICING MORTGAGE-BACKED SECURITIES WITH REPAYMENT RISK UNDER A CIR PROCESS

2012 ◽  
Vol 15 (03) ◽  
pp. 1250021 ◽  
Author(s):  
SEN WU ◽  
LISHANG JIANG ◽  
JIN LIANG
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Reduced Form ◽  
Partial Differential ◽  
Mortgage Backed Securities ◽  
Path Dependent ◽  
Cir Process ◽  
Pass Through

Under a reduced-form framework, we establish models for pricing mortgage-backed securities with prepayment risk by introducing a stochastic prepayment factor. In the zero-default scenario, the pricing pass-through securities and sequential-pay collateralized mortgage obligation structures are considered. To solve the problems, we introduce a path-dependent variable, from which partial differential equation problems are obtained when the prepayment rate is modeled by a CIR process. Numerical solution to the pricing problem is obtained by developing an explicit characteristics difference scheme.

scholarly journals A Stable Difference Scheme for a Third-Order Partial Differential Equation

2018 ◽  
Vol 64 (1) ◽  
pp. 1-19 ◽  
Author(s):  
A Ashyralyev ◽  
Kh Belakroum
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Approximate Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Difference Scheme ◽  
Nonlocal Boundary ◽  
Order Partial Differential Equation ◽  
Third Order ◽  
Partial Differential

The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space H with a self-adjoint positive definite operator A is considered. A stable three-step difference scheme for the approximate solution of the problem is presented. The main theorem on stability of this difference scheme is established. In applications, the stability estimates for the solution of difference schemes of the approximate solution of three nonlocal boundary value problems for third order partial differential equations are obtained. Numerical results for oneand two-dimensional third order partial differential equations are provided.

A Numerical Computational Method for Image Zooming

2014 ◽  
Vol 543-547 ◽  
pp. 2300-2303
Author(s):  
Li Hua Sun ◽  
En Liang Zhao ◽  
Feng Ying Wang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Difference Scheme ◽  
Diffusion Model ◽  
Anisotropic Diffusion ◽  
Integral Method ◽  
Computational Method ◽  
Partial Differential ◽  
The Difference

In this paper, we make use of the integral method which is commonly used in partial differential equations to get the difference scheme on five points, and then construct an anisotropic diffusion model based on the partial differential equation. The numerical experimental results show the diffusion model can effectively magnify the image, and can keep the edge character and details of the image. It is proved that the numerical computational method proposed for solving the model is very effective.

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