Abstract
Solving partial differential equations with discontinuous solutions is an important challenging problem in numerical analysis. To this end, there are some methods such as finite volume method, discontinuous Galerkin approach and particle technique that are able to solve these problems. In the current paper, the moving Kriging element-free Galerkin method has been combined with the variational multiscale algorithm to obtain acceptable and high-resolution solutions. For testing this technique, we select some PDEs with discontinuous solution such as Burgers’, Sod’s shock tube, advection–reaction–diffusion, Kuramoto–Sivashinsky, Boussinesq and shallow water equations. First, we obtain a time-discrete scheme by approximating time derivative via finite difference technique. Then we introduce the moving Kriging interpolation and also obtain their shape functions. We use the element-free Galerkin method for approximating the spatial derivatives. This method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the classical element-free Galerkin method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus, we employ the shape functions of moving Kriging interpolation and radial point interpolation technique which have the mentioned property. Also, in the element-free Galekin method, we do not use any triangular, quadrangular or other type of meshes. The element-free Galerkin method is a global method while finite element method is a local one. This technique employs a background mesh for integration which makes it different from the truly mesh procedures. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed schemes.
Similar content being viewed by others
References
Abedian R, Adibi H, Dehghan M (2014) A high-order symmetrical weighted hybrid ENO-flux limiter scheme for hyperbolic conservation laws. Comput Phys Commun 185:106–127
Abedian R, Adibi H, Dehghan M (2013) A high-order weighted essentially non-oscillatory (WENO) finite difference scheme for nonlinear degenerate parabolic equations. Comput Phys Commun 184:1874–1888
Avesani D, Dumbser M, Bellin A (2014) A new class of moving-least-squares WENO-SPH schemes. J Comput Phys 270:278–299
Belytschko T, Lu YY, Gu L (1994) Element free Galerkin methods. Int J Numer Methods Eng 37:229–256
Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139:3–47
Bui TQ, Nguyen MN (2011) A moving Kriging interpolation-based meshfree method for free vibration analysis of Kirchhoff plates. Comput Struct 89:380–394
Bui TQ, Zhang C (2011) Moving Kriging interpolation-based meshfree method for dynamic analysis of structures. Proc Appl Math Mech 11:197–198
Bui TQ, Nguyen MN, Zhang C (2011) A moving Kriging interpolation-based element-free Galerkin method for structural dynamic analysis. Comput Methods Appl Mech Engin 200:1354–1366
Bratsos AG (1998) The solution of the Boussinesq equation using the method of lines. Comput Methods Appl Mech Eng 157:33–44
Bratsos AG (2010) A fourth-order numerical scheme for solving the modified Burgers equation. Comput Math Appl 60:1393–1400
Chen L, Liew KM (2011) A local Petrov–Galerkin approach with moving Kriging interpolation for solving transient heat conduction problems. Comput Mech 47:455–467
Cheng Y, Bai FN, Peng MJ (2014) A novel interpolating element free Galerkin (IEFG) method for two-dimensional elastoplasticity. Appl Math Model 38:5187–5197
Cheng Y, Li J (2005) A meshless method with complex variables for elasticity. Acta Phys Sin 54:4463–4471
Cheng Y, Peng M (2005) Boundary element free method for elastodynamics. Sci China G 48:641–657
Chung HJ, Belytschko T (1998) An error estimate in the EFG method. Comput Mech 21:91–100
Claina S, Rochette D (2009) First- and second-order finite volume methods for the one-dimensional nonconservative Euler system. J Comput Phys 228:8214–8248
Dai KY, Liu GR, Lim KM, Gu YT (2003) Comparison between the radial point interpolation and the Kriging interpolation used in meshfree methods. Comput Mech 32:60–70
Dai BD, Cheng J, Zheng BJ (2013) Numerical solution of transient heat conduction problems using improved meshless local Petrov–Galerkin method. Appl Math Comput 219:10044–10052
Dai BD, Cheng J, Zheng BJ (2013) A moving Kriging interpolation-based meshless local Petrov–Galerkin method for elastodynamic analysis. Int J Appl Mech 5(1):1350011–1350021
Daǧ İ, Canivar A, Ṣahin A (2011) Taylor–Galerkin and Taylor-collocation methods for the numerical solutions of Burgers equation using B-splines. Commun Nonlinear Sci Numer Simul 16:2696–2708
Dehghan M (2004) Weighted finite difference techniques for the one-dimensional advection-diffusion equation. Appl Math Comput 147:307–319
Dehghan M, Salehi R (2012) A meshless based numerical technique for traveling solitary wave solution of Boussinesq equation. Appl Math Model 36:1939–1956
Dehghan M (2007) Time-splitting procedures for the solution of the two-dimensional transport equation. Kybernetes 36(5):791–805
Dehghan M (2004) Numerical solution of the three-dimensional advection-diffusion equation. Appl Math Comput 150(1):5–19
Dhawan S, Bhowmik SK, Kumar S (2015) Galerkin-least square B-spline approach toward advection-diffusion equation. Appl Math Comput 261:128–140
Gao P (2015) A new global Carleman estimate for the one-dimensional Kuramoto–Sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem. Nonlinear Anal: Theory, Methods Appl 117:133–147
Gu L (2003) Moving Kriging interpolation and element-free Galerkin method. Int J Numer Methods Eng 56:1–11
Gu YT, Zhuang P, Liu F (2010) An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation. Comput Model Eng Sci (CMES) 56:303–334
Gu YT, Wang QX, Lam KY (2007) A meshless local Kriging method for large deformation analyses. Comput Methods Appl Mech Eng 196:1673–1684
Gu YT, Liu GR (2001) A local point interpolation method for static and dynamic analysis of thin beams. Comput Methods Appl Mech Eng 190:5515–5528
Gu YT, Liu GR (2002) A boundary point interpolation method for stress analysis of solids. Comput Mech 28:47–54
Gu YT, Wang W, Zhang LC, Feng XQ (2011) An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields. Eng Fract Mech 78:175–190
Greenough JA, Rider WJ (2004) A quantitative comparison of numerical methods for the compressible Euler equations: fifth-order WENO and piecewise-linear Godunov. J Comput Phys 196:259–281
Hauke G, Garćia-Olivares A (2001) Variational subgrid scale formulations for the advection-diffusion-reaction equation. Comput Methods Appl Mech Eng 190:6847–6865
Huang W, Russell RD (2010) Adaptive moving mesh methods. Springer, Berlin
Huang W, Zheng L, Zhan X (2002) Adaptive moving mesh methods for simulating onedimensional groundwater problems with sharp moving fronts. Int J Numer Methods Eng 54:1579–1603
Jafari H, Borhanifar A, Karimi SA (2009) New solitary wave solutions for the bad Boussinesq and good Boussinesq equations. Numer Methods Partial Differ Equ 25:1231–1237
Jia X, Zeng F, Gu Y (2013) Semi-analytical solutions to one-dimensional advection-diffusion equations with variable diffusion coefficient and variable flow velocity. Appl Math Comput 221:268–281
Jiwari R (2015) A hybrid numerical scheme for the numerical solution of the Burgers equation. Comput Phys Commun 188:59–67
Khan LA, Liu PL-F (1995) An operator splitting algorithm for coupled one-dimensional advection-diffusion-reaction equations. Comput Methods Appl Mech Eng 127:181–201
Khater AH, Temsah RS (2008) Numerical solutions of the generalized Kuramoto–Sivashinsky equation by Chebyshev spectral collocation methods. Comput Math Appl 56:1465–1472
Krongauz Y, Belytschko T (1998) EFG approximation with discontinuous derivatives. Int J Numer Methods Eng 41:1215–1233
Kumar A, Jaiswal DK, Kumar N (2010) Analytical solutions to one-dimensional advection-diffusion equation with variable coefficients in semi-infinite media. J Hydrol 380:330–337
Lakestani M, Dehghan M (2012) Numerical solutions of the generalized Kuramoto–Sivashinsky equation using B-spline functions. Appl Math Model 36:605–617
Lai H, Ma CF (2009) Lattice Boltzmann method for the generalized Kuramoto–Sivashinsky equation. Physica A 388:1405–1412
Lam KY, Wang QX, Li H (2004) A novel meshless approach Local Kriging (LoKriging) method with two-dimensional structural analysis. Comput Mech 33:235–244
Lancaster P, Salkauskas K (1981) Surfaces generated by moving least squares methods. Math Comput 37:141–158
Lee CK, Zhou CE (2004) On error estimation and adaptive refinement for element free Galerkin method Part I: stress recovery and a posteriori error estimation. Comput Struct 82:413–428
Lee CK, Zhou CE (2004) On error estimation and adaptive refinement for element free Galerkin method Part II: adaptive refinement. Comput Struct 82:429–443
LeVeque R (2004) Finite volume methods for hyperbolic problems, Cambridge texts in applied mathematics. Cambridge University Press, Cambridge
Li H, Wang QX, Lam KY (2004) Development of a novel meshless Local Kriging (LoKriging) method for structural dynamic analysis. Comput Methods Appl Mech Eng 193:2599–2619
Li XG, Dai BD, Wang LH (2010) A moving Kriging interpolation-based boundary node method for two-dimensional potential problems. Chin Phys B 19(12):120202–120207
Liu B (2009) An error analysis of a finite element method for a system of nonlinear advection-diffusion-reaction equations. Appl Numer Math 59:1947–1959
Mittal RC, Arora G (2010) Quintic B-spline collocation method for numerical solution of the Kuramoto–Sivashinsky equation. Commun Nonlinear Sci Numer Simul 15:2798–2808
Manoranjan VS, Mitchell AR, Ll J (1984) Morris, numerical solutions of the good Boussinesq equation. SIAM J Sci Stat Comput 5(4):946–957
Mohammadi R (2013) B-spline collocation algorithm for numerical solution of the generalized Burger’s–Huxley equation. Numer Methods Partial Differ Equ 29:1173–1191
Mohebbi A, Dehghan M (2010) High-order compact solution of the one-dimensional heat and advection-diffusion equations. Appl Math Model 34:3071–3084
Mohebbi A, Asgari Z (2011) Efficient numerical algorithms for the solution of “good” Boussinesq equation in water wave propagation. Comput Phys Commun 182:2464–2470
Montecinos GI, Toro EF (2014) Reformulations for general advection-diffusion-reaction equations and locally implicit ADER schemes. J Comput Phys 275:415–442
Seydaoǧlu M, Erdoǧan U, Öziṣ T (2016) Numerical solution of the Burgers equation with high order splitting methods. J Comput Appl Math 291:410–421
Shiue MC (2013) An initial boundary value problem for one-dimensional shallow water magnetohydrodynamics in the solar tachocline. Nonlinear Anal: Theory, Methods Appl 76:215–228
Sod GA (1978) A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J Comput Phys 27:1–31
Shokri A, Dehghan M (2010) A Not-a-Knot meshless method using radial basis functions and predictor-corrector scheme to the numerical solution of improved Boussinesq equation. Comput Phys Commun 181:1990–2000
Ponthot JP, Belytschko T (1998) Arbitrary Lagrangian–Eulerian formulation for element free Galerkin method. Comput Methods Appl Mech Eng 152:19–46
Rademacher JDM, Wattenberg R (2006) Viscous shocks in the destabilized Kuramoto–Sivashinsky. J Comput Nonlinear Dyn 1:336–347
Ren H, Cheng Y (2012) The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems. Eng Anal Bound Elem 36:873–880
Tongsuk P, Kanok-Nukulchai W (2004) Further investigation of element free Galerkin method using moving Kriging interpolation. Int J Comput Methods 01:345–365
Uddin M, Haq S, Islam S (2009) A mesh-free numerical method for solution of the family of Kuramoto–Sivashinsky equations. Appl Math Comput 212:458–469
Wang JF, Sun FX, Cheng YM, Huang AX (2014) Error estimates for the interpolating moving least-squares method. Appl Math Comput 245:321–342
Wazwaz AM (2007) Multiple-front solutions for the Burgers equation and the coupled Burgers equations. Appl Math Comput 190:1198–1206
Wazwaz AM (2004) An analytical study of compacton solutions for variants of Kuramoto–Sivashinsky equation. Appl Math Comput 148:571–585
Wazwaz AM (2011) N-soliton solutions for shallow water waves equations in (1 + 1) and (2 + 1) dimensions. Appl Math Comput 217:8840–8845
Zheng B, Dai BD (2011) A meshless local moving Kriging method for two-dimensional solids. Appl Math Comput 218:563–573
Zhang Z, Hao SY, Liew KM, Cheng YM (2013) The improved element-free Galerkin method for two-dimensional elastodynamics problems. Eng Anal Boundary Elem 37:1576–1584
Zhang LW, Deng YJ, Liew KM (2014) An improved element-free Galerkin method for numerical modeling of the biological population problems. Eng Anal Bound Elem 40:181–188
Zhang LW, Deng YJ, Liew KM, Cheng YM (2014) The improved complex variable element free Galerkin method for two-dimensional Schrödinger equation. Comput Math Appl 68(10):1093–1106
Zhang Z, Liew KM, Cheng Y (2008) Coupling of the improved element-free Galerkin and boundary element methods for two-dimensional elasticity problems. Eng Anal Bound Elem 32:100–107
Zhang Z, Liew KM, Cheng Y, Lee YY (2008) Analyzing 2D fracture problems with the improved element free Galerkin method. Eng Anal Bound Elem 32:241–250
Zhang L, Ouyang J, Wang X, Zhang X (2010) Variational multiscale element-free Galerkin method for 2D Burgers equation. J Comput Phys 229:7147–7161
Zhang L, Ouyang J, Zhang X, Zhang W (2008) On a multiscale element free Galerkin method for the Stokes problem. Appl Math Comput 203:745–753
Zhang L, Ouyang J, Jiang T, Ruan C (2011) Variational multiscale element free Galerkin method for the water wave problems. J Comput Phys 230:5045–5060
Zhang L, Ouyang J, Zhang X (2013) The variational multiscale element free Galerkin method for MHD flows at high Hartmann numbers. Comput Phys Commun 184:1106–1118
Zheng B, Dai B (2011) A meshless local moving Kriging method for two-dimensional solids. Appl Math Comput 218:563–573
Zhao S, Ovadia J, Liu X, Zhang YT, Nie Q (2011) Operator splitting implicit integration factor methods for stiff reaction-diffusion-advection systems. J Comput Phys 230:5996–6009
Zhu P, Zhang LW, Liew KM (2014) Geometrically nonlinear thermo-mechanical analysis of moderately thick functionally graded plates using a local Petrov–Galerkin approach with moving Kriging interpolation. Compos Struct 107:298–314
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Pascal Frey.
Rights and permissions
About this article
Cite this article
Dehghan, M., Abbaszadeh, M. Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some partial differential equations with discontinuous solutions. Comp. Appl. Math. 37, 3869–3905 (2018). https://doi.org/10.1007/s40314-017-0546-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-017-0546-6
Keywords
- Variational multiscale element-free Galerkin (EFG)
- Moving Kriging interpolation
- Burger’s and Sod’s shock tube equations
- Advection–reaction–diffusion and Kuramoto–Sivashinsky equations
- Boussinesq and shallow water equations