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Slope limiters for radial basis functions applied to conservation laws with discontinuous flux function. (English) Zbl 1403.76137
Summary: We present slope limiters in meshless radial basis functions for solving nonlinear equations of conservation laws with flux function that depends on discontinuous coefficients. The method is based on the local collocation formulation and does not require either generation of a grid or evaluation of an integral. Upwind techniques are used to allocate collocation points within the characteristic solutions and different slope limiter functions are investigated. The main advantages of this approach are neither mesh generations nor Riemann problem solvers are required during the solution process. Numerical results are shown for several test examples including models on vehicular traffic and two-phase flows. The main focus is to examine the performance of the proposed meshless method for shock-capturing property in conservation laws with discontinuous flux function. The obtained results demonstrate its ability to capture the main solution features.

76M22 Spectral methods applied to problems in fluid mechanics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L45 Initial value problems for first-order hyperbolic systems
35L65 Hyperbolic conservation laws
76A99 Foundations, constitutive equations, rheology, hydrodynamical models of non-fluid phenomena
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[1] Atluri, S. N.; Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput Mech, 22, 117-127, (1998) · Zbl 0932.76067
[2] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methodsan overview and recent developments, Comput Methods Appl Mech Eng, 139, 3-47, (1996) · Zbl 0891.73075
[3] Benkhaldoun, F.; Sari, S.; Seaid, M., A family of finite volume eulerian-Lagrangian methods for two-dimensional conservation laws, J Comput Appl Math, 285, 181-202, (2015) · Zbl 1315.65076
[4] Buhamman, M., Radial basis function: theory and implementations, (2003), Cambridge University Press, New York
[5] Buhmann, M.; Dinew, S.; Larsson, E., A note on radial basis function interpolant limits, IMA J Numer Anal, 30, 543-554, (2010) · Zbl 1201.65017
[6] Bürger, R.; Karlsen, K. H., Conservation laws with discontinuous fluxa short introduction, J Eng Math, 60, 241-247, (2008) · Zbl 1138.35365
[7] Chan, Y. L.; Shen, L. H.; Wu, C. T.; Young, D. L., A novel upwind based local radial basis function differential quadrature method for convection-dominated flows, Comput Fluids, 89, 157-166, (2014) · Zbl 1391.76529
[8] Diehl, S., A conservation law with point source and discontinuous flux function, SIAM J Math Anal, 56, 388-419, (1996) · Zbl 0849.35142
[9] Franke, R., Scattered data interpolationtests of some methods, Math Comput, 48, 181-200, (1979)
[10] Gimse, T.; Risebro, N. H., Solution of Cauchy problem for a conservation law with a discontinuous flux function, SIAM J Math Anal, 23, 635-648, (1992) · Zbl 0776.35034
[11] Golberg, M. A.; Chen, C. S., The theory of radial basis function applied to the BEM for inhomogeneous partial differential equations, Bound Elem Commun, 5, 57-61, (1994)
[12] Isaacson, E.; Temple, B., Analysis of a singular hyperbolic system of conservation laws, J Diff Equ, 65, 250-268, (1986) · Zbl 0612.35085
[13] Islama, S.; Sarlera, B.; Vertnikb, R.; Kosec, G., Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled burgers׳ equations, J Appl Math Model, 67, 1148-1160, (2013)
[14] Kansa, E. J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics-I surface approximations and partial derivative estimates, Comput Math Appl, 19, 127-145, (1990) · Zbl 0692.76003
[15] Kansa, E. J.; Power, H.; Fasshauer, G. E.; Ling, L., A volumetric integral radial basis function method for time-dependent partial differential equations I. formulation, Eng Anal Bound Elem, 28, 1191-1206, (2004) · Zbl 1159.76363
[16] Khoshfetrat, A.; Abedini, M. J., Numerical modeling of long waves in shallow water using lrbf-DQ and hybrid DQ/LRBF-DQ, Ocean Model, 65, 1-10, (2013)
[17] Krisnamachari, S. V.; Hayes, L. J.; Russel, T. F., A finite element alternating-direction method combined with a modified method of characteristics for convection-diffusion problems, SIAM J Numer Anal, 26, 1462-1473, (1989) · Zbl 0693.65061
[18] Lighthill MJ, Whitham JB. On kinematic waves: II. A theory of traffic flow on long crowded roads. Proc R Soc Lond Ser A 1955;229:317-45. · Zbl 0064.20906
[19] Micchelli, C. A., Interpolation of scattered datadistance matrices and conditionally positive definite functions, Constr Approx, 2, 11-22, (1986) · Zbl 0625.41005
[20] Morton, K. W., Numerical solution of convection-diffusion problems, (1996), Chapman & Hall London · Zbl 0861.65070
[21] Powell, M. J.D., The theory of radial basis function approximation in 1990, in advances in numerical analysis, (Light, W., Wavelets, subdivision algorithms and radial functions, II, (1992), Oxford University Press UK), 105-210 · Zbl 0787.65005
[22] LeVeque Randall, J., Numerical methods for conservation laws, Lectures in mathematics, (1992), ETH Zürich, Zürich · Zbl 0847.65053
[23] Sanyasiraju, Y. V.S. S.; Chandhini, G., Local radial basis function based gridfree scheme for unsteady incompressible viscous flows, J Comput Phys, 227, 8922-8948, (2008) · Zbl 1146.76045
[24] Sarra, S. A., A local radial basis function method for advection-diffusion-reaction equations on complexly shaped domains, J Appl Math Comput, 218, 9853-9865, (2012) · Zbl 1245.65144
[25] Seaid, M., Stable numerical methods for conservation laws with discontinuous flux function, Appl Math Comput, 175, 383-400, (2006) · Zbl 1088.65080
[26] Shu, C., An upwind local RBF-DQ method for simulation of inviscid compressible flows, Comput Methods Appl Mech Eng, 194, 2001-2017, (2005) · Zbl 1093.76052
[27] Shu, C. W., Total variation diminishing time discretizations, SIAM J Sci Stat Comput, 9, 1073-1084, (1988) · Zbl 0662.65081
[28] Towers, J. D., Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J Math Anal, 38, 681-698, (2000) · Zbl 0972.65060
[29] Wendland, H., Scattered data approximation, (2005), Cambridge University Press, Cambridge · Zbl 1075.65021
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