The MacCormack Method – Historical Perspective

Ching, Mao Hung; Deiwert, George S.; Inouye, Mamoru

doi:10.1142/9789812810793_0003

Cited by 2 publications

(3 citation statements)

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“…This approach eliminates the time derivatives, however, it introduces the mixed time and space derivatives which also need removal . Then the modified differential equation's final form (the so‐called Π‐form of the first differential approximation) for the Lax‐Wendroff difference scheme (2) is (using Warming and Hyett's symbols; see [7, p. 166, eq. (3.1)]):

\frac{\partial φ}{\partial t} = {false\sum}_{p = 0}^{3} μ (2 p + 1) \frac{\partial^{2 p + 1} φ}{\partial x^{2 p + 1}} + {false\sum}_{p = 1}^{4} μ (2 p) \frac{\partial^{2 p} φ}{\partial x^{2 p}}

where μ (3), μ (5) and μ (7) are the coefficients which are responsible for the dispersion (phase shift errors) and μ (2), μ (4), μ (6) and μ (8) are the coefficients which are responsible for the dissipation (amplitude errors).…”

Section: The Lax‐wendroff Scheme Modified Differential Equationmentioning

confidence: 99%

“…Thus, the eighth‐order Lax‐Wendroff modified differential equation (3) can be rewritten as (compare to [7, p. 164, eq. (1.6)]):

\begin{array}{l} \frac{\partial φ}{\partial t} + c \frac{\partial φ}{\partial x} & = - \frac{{italicch}^{2}}{6} ((), 1 - λ^{2}) \frac{\partial^{3} φ}{\partial x^{3}} - \frac{{italicch}^{3}}{8} ((), λ - λ^{3}) \frac{\partial^{4} φ}{\partial x^{4}} - \frac{{italicch}^{4}}{120} ((), 1 + 5 λ^{2} - 6 λ^{4}) \frac{\partial^{5} φ}{\partial x^{5}} \\ - \frac{{italicch}^{5}}{48} ((), λ - λ^{3}) \frac{\partial^{6} φ}{\partial x^{6}} - \frac{{italicch}^{6}}{5, 040} ((), 1 + 119 λ^{2} - 210 λ^{4} + 90 λ^{6}) \frac{\partial^{7} φ}{\partial x^{7}} \\ - \frac{{italicch}^{7}}{640} ((), λ + 9 λ^{3} - 20 λ^{5} + 10 λ^{7}) \frac{\partial^{8} φ}{\partial x^{8}} + \dots \end{array}

…”

Section: The Lax‐wendroff Scheme Modified Differential Equationmentioning

confidence: 99%

“…The explicit difference schemes for the constant‐wind‐speed advection equation, especially the Lax‐Wendroff second‐order method, have been commonly dealt with in the recent 50 years—there are many books and papers easily found in the literature concerning the numerical methods for hyperbolic partial differential equations. The history of the MacCormack difference method application in computational fluid dynamics problems may be found in .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

New approach to the Lax‐Wendroff modified differential equation for linear and nonlinear advection

Winnicki

Jasiński

Pietrek

2019

Numerical Methods Partial

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The paper presents an enhanced analysis of the Lax-Wendroff difference scheme-up to the eighth-order with respect to time and space derivatives-of the modified-partial differential equation (MDE) of the constant-wind-speed advection equation. The modified equation has been so far derived mainly as a fourth-order equation. The Π-form of the first differential approximation (differential approximation or equivalent equation) derived by expressing the time derivatives in terms of the space derivatives is used for presenting the MDE. The obtained coefficients at higher order derivatives are analyzed for indications of the character of the dissipative and dispersive errors. The authors included a part of the stencil applied for determining the modified differential equation up to the eighth-order of the analyzed modified differential equation for the second-order Lax-Wendroff scheme. Neither the derived coefficients at the space derivatives of order p ∈ (7 -8) in the modified differential equation for the Lax-Wendroff difference scheme nor the results of analyses on the basis of these coefficients of the group velocity, phase shift errors, or dispersive and dissipative features of the scheme have been published. The MDEs for 2 two-step variants of the Lax-Wendroff type difference schemes and the MacCormack This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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