Exact boundary conditions for the initial value problem of convex conservation laws

Teng, Zhen-Huan

doi:10.1016/j.jcp.2010.01.028

Cited by 2 publications

(2 citation statements)

References 24 publications

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“…As noted in the classic 1982 paper by Fletcher [26], Burgers' equation is often used as a test case for numerical methods to illustrate accuracy and convergence of a particular scheme. This is still true today as illustrated by the recent references (see [23,32,35,38,44,48] and the references therein). It is interesting to note that although Burgers' equation has served as a great test case for numerous numerical algorithms, almost all papers along this line use Burgers' equation with Dirichlet boundary conditions.…”

Section: Introductionmentioning

confidence: 88%

Numerical Approximations of the Dynamical System Generated by Burgers’ Equation with Neumann–Dirichlet Boundary Conditions

Allen¹,

Burns²,

Gilliam³

2013

ESAIM: M2AN

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Using Burgers' equation with mixed Neumann-Dirichlet boundary conditions, we highlight a problem that can arise in the numerical approximation of nonlinear dynamical systems on computers with a finite precision floating point number system. We describe the dynamical system generated by Burgers' equation with mixed boundary conditions, summarize some of its properties and analyze the equilibrium states for finite dimensional dynamical systems that are generated by numerical approximations of this system. It is important to note that there are two fundamental differences between Burgers' equation with mixed Neumann-Dirichlet boundary conditions and Burgers' equation with both Dirichlet boundary conditions. First, Burgers' equation with homogenous mixed boundary conditions on a finite interval cannot be linearized by the Cole-Hopf transformation. Thus, on finite intervals Burgers' equation with a homogenous Neumann boundary condition is truly nonlinear. Second, the nonlinear term in Burgers' equation with a homogenous Neumann boundary condition is not conservative. This structure plays a key role in understanding the complex dynamics generated by Burgers' equation with a Neumann boundary condition and how this structure impacts numerical approximations. The key point is that, regardless of the particular numerical scheme, finite precision arithmetic will always lead to numerically generated equilibrium states that do not correspond to equilibrium states of the Burgers' equation. In this paper we establish the existence and stability properties of these numerical stationary solutions and employ a bifurcation analysis to provide a detailed mathematical explanation of why numerical schemes fail to capture the correct asymptotic dynamics. We extend the results in [E. Allen,

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Section: Introductionmentioning

confidence: 88%

Numerical Approximations of the Dynamical System Generated by Burgers’ Equation with Neumann–Dirichlet Boundary Conditions

Allen¹,

Burns²,

Gilliam³

2013

ESAIM: M2AN

View full text Add to dashboard Cite

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“…e generalised artificial boundaries are classified into global artificial boundaries and local artificial boundaries. Global artificial boundaries are mainly characterised by coupled motion of all boundary nodes in space and time and satisfaction of all field equations and mathematical and physical conditions in the infinite domain, which can accurately simulate the infinite domain [9][10][11]. In the analysis of large-scale complex wave motion, especially nonlinear generalised structures, global artificial boundaries impose an onerous computational burden.…”

Section: Introductionmentioning

confidence: 99%

An Arc-Consistent Viscous-Spring Artificial Boundary for Numerical Analysis of Seismic Response of Underground Structures

Yin²,

Wang

et al. 2022

Shock and Vibration

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A new arc-consistent viscous-spring artificial boundary (ACVAB) was proposed by changing a traditional flat artificial boundary based on the theory of viscous-spring artificial boundaries. Through examples, the concept underpinning the establishment and specific setting of the boundary in the finite element software were described. Through comparison with other commonly used artificial boundaries in an example for near-field wave analysis using the two-dimensional (2D) half-space model, the reliability of the ACVAB was verified. Furthermore, the ACVAB was used in the numerical analysis of the effects of an earthquake on underground structures. The results were compared with the shaking table test results on underground structures. On this basis, the applicability of the ACVAB to a numerical model of seismic response of underground structures was evaluated. The results show that the boundary is superior to common viscous-spring boundaries in terms of accuracy and stability, and therefore, it can be used to evaluate radiation damping effects of seismic response of underground structures and is easier to use.

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Exact boundary conditions for the initial value problem of convex conservation laws

Cited by 2 publications

References 24 publications

Numerical Approximations of the Dynamical System Generated by Burgers’ Equation with Neumann–Dirichlet Boundary Conditions

Numerical Approximations of the Dynamical System Generated by Burgers’ Equation with Neumann–Dirichlet Boundary Conditions

An Arc-Consistent Viscous-Spring Artificial Boundary for Numerical Analysis of Seismic Response of Underground Structures

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