THE EARLY INFLUENCE OF PETER LAX ON COMPUTATIONAL HYDRODYNAMICS AND AN APPLICATION OF LAX-FRIEDRICHS AND#br#
LAX-WENDROFF ON TRIANGULAR GRIDS IN LAGRANGIAN COORDINATES

doi:10.1016/S0252-9602(11)60393-7

Acta mathematica scientia,Series B ›› 2011, Vol. 31 ›› Issue (6): 2195-2202.doi: 10.1016/S0252-9602(11)60393-7

• Articles • Previous Articles Next Articles

THE EARLY INFLUENCE OF PETER LAX ON COMPUTATIONAL HYDRODYNAMICS AND AN APPLICATION OF LAX-FRIEDRICHS AND#br# LAX-WENDROFF ON TRIANGULAR GRIDS IN LAGRANGIAN COORDINATES

Richard Liska¹, Mikhail Shashkov², Burton Wendroff³

1.Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Bˇrehová|7, 115 19 Prague 1, Czech Republic|2.Group XCP-4, MS-F644, Los Alamos National Laboratry, Los Alamos, New Mexico 87545, USA|3.Group T-5, MS-B284, Los Alamos National Laboratry, Los Alamos, New Mexico 87545, USA

Received:2011-07-27 Online:2011-11-20 Published:2011-11-20
Supported by:
This work was performed under the auspices of the National Nuclear Secu-rity Administration of the US Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. The first author has been supported in part by the Czech Science Foundation Grant P205/10/0814 and by the Czech Ministry of Education grants MSM 6840770022 and LC528.

Abstract

Abstract:

We give a brief discussion of some of the contributions of Peter Lax to Com-putational Fluid Dynamics. These include the Lax-Friedrichs and Lax-Wendroff numerical schemes. We also mention his collaboration in the 1983 HLL Riemann solver. We de-velop two-dimensional Lax-Friedrichs and Lax-Wendroff schemes for the Lagrangian form of the Euler equations on triangular grids. We apply a composite scheme that uses a Lax-Friedrichs time step as a dissipative filter after several Lax-Wendroff time steps. Numerical results for Noh’s infinite strength shock problem, the Sedov blast wave problem, and the Saltzman piston problem are presented.

Key words: Lax-Friedrichs, Lax-Wendroff, conservation laws, Lagrangian coordinates, triangular grid

CLC Number:

35L65

Richard Liska, Mikhail Shashkov, Burton Wendroff. THE EARLY INFLUENCE OF PETER LAX ON COMPUTATIONAL HYDRODYNAMICS AND AN APPLICATION OF LAX-FRIEDRICHS AND#br# LAX-WENDROFF ON TRIANGULAR GRIDS IN LAGRANGIAN COORDINATES[J].Acta mathematica scientia,Series B, 2011, 31(6): 2195-2202.

Trendmd

References

[1] Lax P D. Weak solutions of nonlinear hyperbolic equations and their numerical computation. Comm Pure Appl Math, 1954, 7: 159–193

[2] Lax P D, Wendroff B. Systems of conservation laws. Comm Pure Appl Math, 1960, 13: 217–237

[3] Lax P D, Wendroff B. Difference schemes for hyperbolic equations with high order of accuracy. Comm Pure Appl Math, 1964, 17: 381–398

[4] Richtmyer R D. A survey of difference methods for non-steady fluid dynamics. Tech Report NCAR Technical Notes 63-2, National Center for Atmospheric Research, 1962

[5] Harten A, Lax P D, van Leer B. On upstream differencing and Godunov-type schemes for hyperbolic conaervation laws. SIAM Review, 1983, 25(1): 35–61

[6] Loubere R, Shashkov M. A subcell remapping method on staggered polygonal grids for arbitrary lagrangian eulerian methods. J Comp Phys, 2005, 209: 105–138

[7] Maire P H. A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes. J Comp Phys, 2009, 228(7): 2391–2425

[8] Farhat C, Geuzaine P, Grandmont C. The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of the flow problems on the moving grids. J Comp Phys, 2001, 174: 669–694

[9] Noh W F. Errors for calculations of strong shocks using an artificial viscosity and artificial heat flux. J Comp Phys, 1987, 72: 78–120

[10] Sedov L I. Similarity and Dimensional Methods in Mechanics. New York: Academic Press, 1959

[11] Kamm J R, Timmes F X. On efficient generation of numerically robust Sedov solutions. Tech Report LA-UR-07-2849, Los Alamos National Laboratory, 2007

[12] Caramana E J, Burton D E, Shashkov M J, Whalen P P. The construction of compatible hydrodynamics algorithms utilizing conservation of total energy. J Comp Phys, 1998, 146(1): 227–262

[13] Laney C B. Computational Gas Dynamics. Cambridge University Press, 1998

[14] Shashkov M, Wendroff B. A composite scheme for gas dynamics in Lagrangian coordinates. J Comp Phys, 1999, 150: 502–517

[15] Liska R, Shashkov M, Wendroff B. Lagrangian composite schemes on triangular unstructured grids//Kocandrlova M, Kelar V, eds. Mathematical and Computer Modelling in Science and Engineering, Inter-national Conference in honour of the 80th birthday of K. Rektorys. Prague 2003

[16] Toro E F. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. 2nd ed. Springer-Verlag, 1999

[17] Liska R, Wendroff B. Composite schemes for conservation laws. SIAM J Numer Anal, 1998, 35(6): 2250–2271

THE EARLY INFLUENCE OF PETER LAX ON COMPUTATIONAL HYDRODYNAMICS AND AN APPLICATION OF LAX-FRIEDRICHS AND#br# LAX-WENDROFF ON TRIANGULAR GRIDS IN LAGRANGIAN COORDINATES

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Shujun LIU, Fangqi CHEN, Zejun WANG. EXISTENCE OF GLOBAL L^∞ SOLUTIONS TO A GENERALIZED n×n HYPERBOLIC SYSTEM OF LEROUX TYPE [J]. Acta mathematica scientia,Series B, 2018, 38(3): 889-897.
[2]	Corrado MASCIA. TWENTY-EIGHT YEARS WITH “HYPERBOLIC CONSERVATION LAWS WITH RELAXATION” [J]. Acta mathematica scientia,Series B, 2015, 35(4): 807-831.
[3]	Debora AMADORI, Paolo BAITI, Andrea CORLI, Edda DAL SANTO. GLOBAL EXISTENCE OF SOLUTIONS FOR A MULTI-PHASE FLOW: A BUBBLE IN A LIQUID TUBE AND RELATED CASES [J]. Acta mathematica scientia,Series B, 2015, 35(4): 832-854.
[4]	Gaowei CAO, Kai HU, Xiaozhou YANG. FORMULA OF GLOBAL SMOOTH SOLUTION FOR NON-HOMOGENEOUS M-D CONSERVATION LAW WITH UNBOUNDED INITIAL VALUE [J]. Acta mathematica scientia,Series B, 2015, 35(2): 508-526.
[5]	Adil JHANGEER. CONSERVATION LAWS FOR THE (1 + 2)-DIMENSIONAL WAVE EQUATION IN BIOLOGICAL ENVIRONMENT [J]. Acta mathematica scientia,Series B, 2013, 33(5): 1255-1268.
[6]	Stefano Bianchini. SBV REGULARITY OF GENUINELY NONLINEAR HYPERBOLIC SYSTEMS OF CONSERVATION LAWS IN ONE SPACE DIMENSION [J]. Acta mathematica scientia,Series B, 2012, 32(1): 380-388.
[7]	Michael G. Hilgers. NONUNIQUENESS AND SINGULAR RADIAL SOLUTIONS OF SYSTEMS OF CONSERVATION LAWS [J]. Acta mathematica scientia,Series B, 2012, 32(1): 367-379.
[8]	Geng Chen, Robin Young. THE VACUUM IN NONISENTROPIC GAS DYNAMICS [J]. Acta mathematica scientia,Series B, 2012, 32(1): 339-351.
[9]	Rinaldo M. Colombo, Magali L′ecureux-Mercier. NONLOCAL CROWD DYNAMICS MODELS FOR SEVERAL POPULATIONS [J]. Acta mathematica scientia,Series B, 2012, 32(1): 177-196.
[10]	WANG Zhen, ZHANG Hui. FREE BOUNDARY VALUE PROBLEM OF ONE DIMENSIONAL TWO-PHASE LIQUID-GAS MODEL [J]. Acta mathematica scientia,Series B, 2012, 32(1): 413-432.
[11]	SUN Mei-Na. A NOTE ON THE INTERACTIONS OF ELEMENTARY WAVES FOR THE AR TRAFFIC FLOW MODEL WITHOUT VACUUM [J]. Acta mathematica scientia,Series B, 2011, 31(4): 1503-1512.
[12]	WANG Wei-Ke. NONLINEAR EVOLUTION SYSTEMS AND GREEN'S FUNCTION [J]. Acta mathematica scientia,Series B, 2010, 30(6): 2051-2063.
[13]	H. Kim, M. Laforest, D. Yoon. AN ADAPTIVE VERSION OF GLIMM'S SCHEME [J]. Acta mathematica scientia,Series B, 2010, 30(2): 428-446.
[14]	Christian Klingenberg, Knut Waagan. RELAXATION SOLVERS FOR IDEAL MHD EQUATIONS -A REVIEW [J]. Acta mathematica scientia,Series B, 2010, 30(2): 621-632.
[15]	Gui-Qiang G. Chen, Weihua Ruan. A HYPERBOLIC SYSTEM OF CONSERVATION LAWS FOR FLUID FLOWS THROUGH COMPLIANT XISYMMETRIC VESSELS [J]. Acta mathematica scientia,Series B, 2010, 30(2): 391-427.