Results 191 to 200 of about 160,914 (245)
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2019
In fluid dynamics, the classical Riemann problem (Riemann 1860) is an initial-value problem for a set of homogeneous PDEs in which the initial data consist of two constant states forming a discontinuity (Toro 1997, 2001; LeVeque 2002; Guinot 2003) (Fig. 8.1). It is a generalization of the dam break problem (Stoker 1957) described in Chap. 6.
Oscar Castro-Orgaz, Willi H. Hager
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In fluid dynamics, the classical Riemann problem (Riemann 1860) is an initial-value problem for a set of homogeneous PDEs in which the initial data consist of two constant states forming a discontinuity (Toro 1997, 2001; LeVeque 2002; Guinot 2003) (Fig. 8.1). It is a generalization of the dam break problem (Stoker 1957) described in Chap. 6.
Oscar Castro-Orgaz, Willi H. Hager
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SIAM Journal on Mathematical Analysis, 1999
The authors consider the flow of an isothermal gas in a (infinitely) long, thin pipe of constant section, which has one or several kinks in it. They obtain the following model for the flow: \[ \rho_t+ (\rho v)_x= 0,\quad (\rho v)_t+ (\rho v^2+ \rho)_x= -f\sum_i k_i \delta_{x_i} \rho v, \] where \(\delta_{x_i}\) is the Dirac measure located at the ...
Holden, Helge, Risebro, Nils Henrik
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The authors consider the flow of an isothermal gas in a (infinitely) long, thin pipe of constant section, which has one or several kinks in it. They obtain the following model for the flow: \[ \rho_t+ (\rho v)_x= 0,\quad (\rho v)_t+ (\rho v^2+ \rho)_x= -f\sum_i k_i \delta_{x_i} \rho v, \] where \(\delta_{x_i}\) is the Dirac measure located at the ...
Holden, Helge, Risebro, Nils Henrik
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1987
In Chapter I we analyzed the mapping from the functions Ψ(x), to the transition coefficients and discrete spectrum of the auxiliary linear problem. We saw that for both rapidly decreasing and finite density boundary conditions this “change of variables” makes the dynamics quite simple because the time evolution of the transition coefficients ...
Ludwig D. Faddeev, Leon A. Takhtajan
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In Chapter I we analyzed the mapping from the functions Ψ(x), to the transition coefficients and discrete spectrum of the auxiliary linear problem. We saw that for both rapidly decreasing and finite density boundary conditions this “change of variables” makes the dynamics quite simple because the time evolution of the transition coefficients ...
Ludwig D. Faddeev, Leon A. Takhtajan
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The Riemann and Riemann-Hilbert Problems
1987In this chapter the Riemann and the Riemann-Hilbert problems are stated. The first problem is so easy to solve that one might say it is almost obvious. Nevertheless it is discussed here in order to prepare for the exposition of the same question in several variables studied in Chapter 9 which is by no means easy.
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2002
In this chapter, we study the Riemann problem for scalar conservation laws. In Section 1 we discuss several formulations of the entropy condition. Then, in Section 2 we construct the classical entropy solution satisfying, by definition, all of the entropy inequalities; see Theorems 2.1 to 2.4.
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In this chapter, we study the Riemann problem for scalar conservation laws. In Section 1 we discuss several formulations of the entropy condition. Then, in Section 2 we construct the classical entropy solution satisfying, by definition, all of the entropy inequalities; see Theorems 2.1 to 2.4.
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2016
In his famous dissertation from 1851, Bernhard Riemann (1826–1866) formulated the following problem: In a given bounded domain of the complex plane, determine a holomorphic function, if a relation is prescribed between the boundary values of its real part and its imaginary part. In the case of a linear relation this problem was first considered in 1904
Klaus Gürlebeck +2 more
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In his famous dissertation from 1851, Bernhard Riemann (1826–1866) formulated the following problem: In a given bounded domain of the complex plane, determine a holomorphic function, if a relation is prescribed between the boundary values of its real part and its imaginary part. In the case of a linear relation this problem was first considered in 1904
Klaus Gürlebeck +2 more
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The Riemann-Hilbert Problem on the Riemann Surface
2019Here we reduce Eq. ( 15.11) to the Riemann-Hilbert problem. Let us recall that the function \(\hat v_{21}\) is meromorphic in \( \varPi _{-2\varPhi }^\pi \) by ( 15.9) and Lemma 15.2. Consider the strip $$\displaystyle W:=\varPi _{\pi -2\varPhi }^\pi .$$
Alexander Komech, Anatoli Merzon
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2013
This chapter is devoted to study the Riemann problems for scalar conservation laws in one space dimension. In particular, we introduce the definitions of entropy and non-entropy shock waves, rarefaction waves and contact discontinuities by first considering genuinely nonlinear characteristic fields, linearly degenerate characteristic fields and then ...
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This chapter is devoted to study the Riemann problems for scalar conservation laws in one space dimension. In particular, we introduce the definitions of entropy and non-entropy shock waves, rarefaction waves and contact discontinuities by first considering genuinely nonlinear characteristic fields, linearly degenerate characteristic fields and then ...
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2000
Abstract Let γ ⊆ ℝn be an open set and let f : γ ↦ ℝn be a smooth vector field. The Riemann problem for the system of conservation laws consists in finding a weak solution of (5.1) with piecewise constant initial data of the ...
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Abstract Let γ ⊆ ℝn be an open set and let f : γ ↦ ℝn be a smooth vector field. The Riemann problem for the system of conservation laws consists in finding a weak solution of (5.1) with piecewise constant initial data of the ...
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