## New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection-Diffusion Equations (2000)

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Venue: | J. Comput. Phys |

Citations: | 141 - 15 self |

### BibTeX

@ARTICLE{Kurganov00newhigh-resolution,

author = {A. Kurganov and E. Tadmor},

title = {New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection-Diffusion Equations},

journal = {J. Comput. Phys},

year = {2000},

volume = {160},

pages = {241--282}

}

### Years of Citing Articles

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### Abstract

this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order )).In particular, our new central schemes maintain their high-resolution independent of O(1/#t ), and letting #t 0, they admit a particularly simple semi-discrete formulation

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Citation Context ... (x j := j�x, t n := n�t). Compared with the canonical first-order upwind scheme of Godunov [11], the central LxF scheme has the advantage of simplicity, since no (approximate) Riemann solvers, e.=-=g., [42]-=-, are involved in its construction. The main disadvantage of the LxF scheme, however, lies in its large numerical dissipation, which prevents sharp resolution of shock discontinuities and rarefaction ... |

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Citation Context ...[30]; it is a special case of the family of first-order Godunov-type scheme introduced in [15] (based on a symmetric approximate Rieman solver) and it coincides with the so-called local LxF scheme in =-=[45].-=- It should be noted, however, that although this scheme is similar to the LxF scheme, (2.7), its numerical viscosity coefficient, Qn j+1/2 = λanj+1/2 , is always smaller than the corresponding LxF on... |

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Towards the Ultimate Conservative Difference Scheme V. A Second Order Sequel to Godunov’s Method
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Citation Context ... n 1 � � n n − f u j+1 − f u j−1 + λa j+1/2 u j+1 − u 2 2 n� � n n j − λa j−1/2 u j − u n �� j−1 , (3.10) where an j+1/2 are the maximal local speeds. This scheme was ori=-=ginally attributed to Rusanov [30]-=-; it is a special case of the family of first-order Godunov-type scheme introduced in [15] (based on a symmetric approximate Rieman solver) and it coincides with the so-called local LxF scheme in [45]... |

307 |
High-Resolution Schemes for Hyperbolic Conservation
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- 1983
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Citation Context ...y property of this piecewise-linear approximation (e.g., decreasing the number of extrema [36, Section 4]) can be satisfied for a wide variety of such scalar reconstructions proposed and discussed in =-=[4, 13, 14, 27, 32, 36, 38, 41]. For example,-=- a scalar TVD reconstruction in (2.3) is obtained via the ubiquitous minmod limiter [13, 31, 41], (ux) n � n u j − u j = minmod n j−1 , �x unj+1 − un � j , (2.4) �x with minmod(a, b) := ... |

219 | Non-oscillation central differencing for hyperbolic conservation laws
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- 1990
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Citation Context ...several other areas. In this work we present new second-order central difference approximations to (1.1) and (1.2). These new schemes can be viewed as modifications of the Nessyahu–Tadmor (NT) schem=-=e [38]. -=-Our schemes enjoy the major advantages of the central schemes oversHIGH-RESOLUTION CENTRAL SCHEMES 243 the upwind ones: first, no Riemann solvers are involved, and second—as a result of being Rieman... |

181 |
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Citation Context ...y property of this piecewise-linear approximation (e.g., decreasing the number of extrema [36, Section 4]) can be satisfied for a wide variety of such scalar reconstructions proposed and discussed in =-=[4, 13, 14, 27, 32, 36, 38, 41]. For example,-=- a scalar TVD reconstruction in (2.3) is obtained via the ubiquitous minmod limiter [13, 31, 41], (ux) n � n u j − u j = minmod n j−1 , �x unj+1 − un � j , (2.4) �x with minmod(a, b) := ... |

158 |
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Citation Context ...�� j−1 , (3.10) where an j+1/2 are the maximal local speeds. This scheme was originally attributed to Rusanov [30]; it is a special case of the family of first-order Godunov-type scheme introduc=-=ed in [15]-=- (based on a symmetric approximate Rieman solver) and it coincides with the so-called local LxF scheme in [45]. It should be noted, however, that although this scheme is similar to the LxF scheme, (2.... |

125 |
Weak Solution of Nonlinear Hyperbolic Equations and their
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Citation Context ...ided; in the particular case of the second-order NT scheme, for example, numerical derivatives of the flux in (2.6) below can be implemented componentwise—consult [18, 36]. In 1954 Lax and Friedrich=-=s [10, 29] introduced the first-order -=-stable central scheme, the celebrated Lax–Friedrichs (LxF) scheme: u n+1 j = un j+1 + un j−1 2 − λ� � � � �� n n f u j+1 − f u j−1 . (2.1) 2 Here, λ := �t/�x is the fixed m... |

120 |
Finite Difference Method for Numerical Computation of Discontinuous Solutions of the Equations of Fluid Dynamics
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Citation Context ...:= �t/�x is the fixed mesh ratio, and un j is an approximate value of u(x = x j, t = t n ) at the grid point (x j := j�x, t n := n�t). Compared with the canonical first-order upwind scheme of =-=Godunov [11]-=-, the central LxF scheme has the advantage of simplicity, since no (approximate) Riemann solvers, e.g., [42], are involved in its construction. The main disadvantage of the LxF scheme, however, lies i... |

115 |
A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws
- Sod
- 1978
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Citation Context ...eported in [5].sby HIGH-RESOLUTION CENTRAL SCHEMES 265 FIG. 6.7. Riemann IVP (6.7). N = 100. We compute the solution to two different Riemann problems: • The first Riemann problem was proposed by So=-=d [46]. The -=-initial data are given �u L = (1, 0, 2.5) T , �u R = (0.125, 0, 0.25) T . The approximations to the density, velocity, and pressure obtained by the FD2 scheme are presented in Figs. 6.9–6.14. FI... |

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Citation Context ...orward Euler scheme is limited to first-order accuracy. It can be used, however, as a building block for higher-order schemes based on Runge–Kutta (RK) or multi-level time differencing. Shu and Oshe=-=r [44, 45] h-=-ave identified a whole family of such schemes, based on convex combinations of forward Euler steps. To this end, we let C[w] denote our spatial recipe (4.17)–(4.19) for central differencing a grid f... |

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Citation Context ...DMOR solvers, characteritic decompositions, etc. This aspect in the context of high-resolution schemes was introduced in the Nessyahu–Tadmor scheme [38] and was extended to twodimensional problems i=-=n [18]-=-. A nonstaggered and hence less dissipative version was presented in [17]. The relaxation scheme introduced in [19] is closely related to these staggered central schemes; in fact, they coicide in the ... |

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Citation Context ...ation of the Jacobian of f can be avoided; in the particular case of the second-order NT scheme, for example, numerical derivatives of the flux in (2.6) below can be implemented componentwise—consul=-=t [18, 36]. In 1954 Lax and Fried-=-richs [10, 29] introduced the first-order stable central scheme, the celebrated Lax–Friedrichs (LxF) scheme: u n+1 j = un j+1 + un j−1 2 − λ� � � � �� n n f u j+1 − f u j−1 . (2... |

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- 1954
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Citation Context ...ided; in the particular case of the second-order NT scheme, for example, numerical derivatives of the flux in (2.6) below can be implemented componentwise—consult [18, 36]. In 1954 Lax and Friedrich=-=s [10, 29] introduced the first-order -=-stable central scheme, the celebrated Lax–Friedrichs (LxF) scheme: u n+1 j = un j+1 + un j−1 2 − λ� � � � �� n n f u j+1 − f u j−1 . (2.1) 2 Here, λ := �t/�x is the fixed m... |

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Citation Context ... f , this is further simplified: a j+1/2(t) := max{| f ′ (u − j+1/2 (t))|, | f ′ (u + j+1/2 (t))|}. (4.11) Proof. The second-order flux in (4.2), H j+1/2(t), can be viewed as a generalized MUSCL=-= flux [40], H j+1/2(t)-=- = H Rus� u + j+1/2 (t), u− j+1/2 (t)� , expressed in terms of the first-order E-flux H Rus = H Rus (uℓ, ur), associated with the firstorder Rusanov scheme (4.8), H Rus (uℓ, ur) := f (uℓ) ... |

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Citation Context ...ar problems. In Figs. 6.5 and 6.6 we present the approximate solutions at the post-shock time T = 2, when the shock is well developed. Second-order behavior is confirmed by the measuring Lip´-errors,=-= [39], -=-which are recorded in Table 6.3. Again, the solution obtained by the FD2 scheme is slightly more accurate than the solution computed by the NT scheme. FIG. 6.6. Problem (6.5)–(6.6). N = 100, T = 2; ... |

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6 |
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- 1999
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Citation Context ... Riemann problem for the Buckley–Leverett equation with and without gravitation. T = 0.2. EXAMPLE 9 (Glacier Growth Model). In this example we consider a one-dimensional model for glacier growth (se=-=e [9, 21]). Le-=-t a glacier of height h(x, t) rest upon a flat mountain. Its evolution is described by the nonhomogenious convection–diffusion equation ht + f (h)x = ε(ν(h)hx)x + S(x, t, h). (6.14) Let ε = 0.01.... |

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2 |
private communication
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- 1994
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Citation Context ...on schemes; we note in passing that the special scalar choice A = ρ(∂f (u)/∂u)I is an O(ε) perturbation of the central scheme discussed in this paper. Other componentwise approaches were present=-=ed in [34]-=-. The CUSP scheme presented in [16] is a semi-discrete scheme which avoids characteristic decompositions. And more recently, Liu and Osher [35] introduced a semi-discrete scheme based on a pointwise f... |

1 |
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Citation Context ... Riemann problem for the Buckley–Leverett equation with and without gravitation. T = 0.2. EXAMPLE 9 (Glacier Growth Model). In this example we consider a one-dimensional model for glacier growth (se=-=e [9, 21]). Le-=-t a glacier of height h(x, t) rest upon a flat mountain. Its evolution is described by the nonhomogenious convection–diffusion equation ht + f (h)x = ε(ν(h)hx)x + S(x, t, h). (6.14) Let ε = 0.01.... |