New optimized meshless block method for solving time-dependent PDEs
Abstract views: 0
/ 
PDF downloads: 0
Keywords:
PDEs;, Discretization;, Optimization;, Hybrid block method;, Mesh-based approaches;, Meshless methods;, RBF-FD;, OMBMAbstract
Partial differential equations (PDEs) provide mathematical models to describe real phenomena such as heat conduction, wave propagation, and many other scientific disciplines. A vast array of methods has been used to solve PDEs. Among them, multistep block methods, along with mesh-based techniques, have been utilized to discretize the time variables and the partial derivatives with respect to the spatial variable in the PDE. However, this approach encounters challenges such as discontinuities, high computational cost, and time demands. In this context, the current work proposes an Optimized Meshless Block Method (OMBM) for solving time-dependent PDEs. The method integrates RBF-FD for spatial discretization with a two-step hybrid block method for time integration. This combination leverages the accuracy and stability of block methods, along with the geometric flexibility and reduced computational cost offered by meshless approaches. The approach is strengthened by a strategic choice of shape parameter, which mitigates the well-known sensitivity issue inherent to RBFs, thereby enhancing the overall robustness and reliability of the numerical solution across varying spatial resolutions. Various test problems are examined, and the simulation results are compared with exact solutions and prior studies to demonstrate the superior performance and accuracy of the proposed approach.
References
Romeo, Giovanni. Elements of numerical mathematical economics with Excel: static and dynamic optimization. Academic Press, 2019. https://doi.org/10.1016/B978-0-12-817648-1.02003-X
Ramos, H., Kaur, A., & Kanwar, V. (2022). Using a cubic B-spline method in conjunction with a one-step optimized hybrid block approach to solve nonlinear partial differential equations. Computational and Applied Mathematics, 41(1). https://doi.org/10.1007/s40314-021-01729-7
Ferreira, L. A., & Mesquita, R. C. (2015). Graphical Interface for Electromagnetic Problem Solving Using Meshless Methods. In Article in Journal of Microwaves Optoelectronics and Electromagnetic Applications (Vol. 14, Issue 1).
Courant, R., Friedrichs, K., & Lewy, H. (1928). Über die partiellen Differenzengleichungen der mathematischen Physik. Mathematische annalen, 100(1), 32-74. https://doi.org/10.1007/BF01448839
Ciarlet, P. G., & Raviart, P. A. (1973). Maximum principle and uniform convergence for the finite element method. Computer methods in applied mechanics and engineering, 2(1), 17-31. https://doi.org/10.1016/0045-7825(73)90019-4
Tian, M. , & C. Z. (1991). Quadratic element generalized differential methods for elliptic equations. Numer. Math. J. Chinese Univ, 13, 99-13.
Costabel, M. (1987). Principles of boundary element methods. Computer Physics Reports, 6(1-6), 243-274. https://doi.org/10.1016/0167-7977(87)90014-1
Bernardi, C., & Maday, Y. (1997). Spectral methods. Handbook of numerical analysis, 5, 209-485. https://doi.org/10.1016/S1570-8659(97)80003-8
Smith, B. F. (1997). Domain Decomposition Methods for Partial Differential Equations (pp. 225-243). https://doi.org/10.1007/978-94-011-5412-3_8
Fasshauer, G. E. (2007). Meshfree approximation methods with MATLAB (Vol. 6). World Scientific. https://doi.org/10.1142/6437
Burkhardt, J., Wu, Y., Computing, Y. Z.-S. J. on S., & 2021, undefined. (2021). A unified meshfree pseudospectral method for solving both classical and fractional PDEs. SIAMJ Burkardt, Y Wu, Y ZhangSIAM Journal on Scientific Computing, 2021. https://doi.org/10.1137/20M1335959
Eason, E. D. (1976). A review of least‐squares methods for solving partial differential equations. International Journal for Numerical Methods in Engineering, 10(5), 1021-1046. https://doi.org/10.1002/nme.1620100505
Nayroles, B., Touzot, G., & Villon, P. (1992). Generalizing the finite element method: Diffuse approximation and diffuse elements. Computational Mechanics, 10(5), https://doi.org/10.1007/BF00364252
Belytschko, T., Lu, Y. Y., & Gu, L. (1994). Element‐free Galerkin methods. International journal for numerical methods in engineering, 37(2), 229-256. https://doi.org/10.1002/nme.1620370205
Duarte, C. A., & Oden, J. T. (1996). An hp adaptive method using clouds. Computer methods in applied mechanics and engineering, 139(1-4), 237-262. https://doi.org/10.1016/S0045-7825(96)01085-7
Melenk, J. M., & Babuška, I. (1996). The partition of unity finite element method: basic theory and applications. Computer methods in applied mechanics and engineering, 139(1-4), 289-314. https://doi.org/10.1016/S0045-7825(96)01087-0
Liu, Jun, S., & Zhang, Y. F. (1995). Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids, 20(8-9), 1081-1106. https://doi.org/10.1002/fld.1650200824
Atluri, S. N., & Zhu, T. (1998). A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational mechanics, 22(2), 117-127. https://doi.org/10.1007/s004660050346
Aluru, N. (2000). A point collocation method based on reproducing kernel approximations. International Journal for Numerical Methods in Engineering, 47(6), 1083-1121. https://doi.org/10.1002/(SICI)1097-0207(20000228)47:6<1083::AID-NME816>3.0.CO;2-N
Kansa, E. J. (1990). Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics-II. In Computers Math. Applic (Vol. 19, Issue 9). https://doi.org/10.1016/0898-1221(90)90271-K
Monaghan, J. J. (2003). Smoothed Particle Hydrodynamics.
Ahmad, I., Siraj-ul-Islam, & Khaliq, A. Q. M. (2017). Local RBF method for multi-dimensional partial differential equations. Computers & Mathematics with Applications, 74(2), 292-324. https://doi.org/10.1016/j.camwa.2017.04.026
Slak, J. (2020). Adaptive RBF-FD method: PhD thesis (Doctoral dissertation, Univerza v Ljubljani, Fakulteta za matematiko in fiziko).
Tillenius, M., Larsson, E., Lehto, E., & Flyer, N. (2015). A scalable RBF-FD method for atmospheric flow. Journal of Computational Physics, 298, 406-422. https://doi.org/10.1016/j.jcp.2015.06.003
Chen, J.-S., Hillman, M., & Chi, S.-W. (2017). Meshfree Methods: Progress Made after 20 Years. Journal of Engineering Mechanics, 143(4), 04017001. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001176
Koushki, M., Jabbari, E., & Ahmadinia, M. (2020). Evaluating RBF methods for solving PDEs using Padua points distribution. Alexandria Engineering Journal, 59(5), 2999-3018. https://doi.org/10.1016/j.aej.2020.04.047
Chen, W., Fu, Z. J., & Chen, C. S. (2014). Recent advances in radial basis function collocation methods (Vol. 768). Berlin: Springer https://doi.org/10.1007/978-3-642-39572-7
G.B. Wright, B. Fornberg, Scattered node compact finite difference-type formulas generated from radial basis functions. J. Comput. Phys. 212(1), 99-123 (2006) https://doi.org/10.1016/j.jcp.2005.05.030
V. Bayona, M. Moscoso, M. Carretero, M. Kindelan, RBF-FD formulas and convergence properties. J. Comput. Phys. 229(22), 8281-8295 (2010) https://doi.org/10.1016/j.jcp.2010.07.008
V. Bayona, M. Moscoso, M. Kindelan, Optimal constant shape parameter for multiquadric based RBF-FD method. J. Comput. Phys. 230(19), 7384-7399 (2011) https://doi.org/10.1016/j.jcp.2011.06.005
Y. Liu, Y.C. Hon, K.M. Liew, A meshfree Hermite-type radial point interpolation method for Kirchhoff plate problems. Int. J. Numer. Meth. Eng. 66(7), 1153-1178 (2006) https://doi.org/10.1002/nme.1587
C. Shu, H. Ding, K.S. Yeo, Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 192, 941-954 (2003) https://doi.org/10.1016/S0045-7825(02)00618-7
S. Quan, Local RBF-based differential quadrature collocation method for the boundary layer problems. Eng. Anal. Boundary Elem. 34(3), 213-228 (2010) https://doi.org/10.1016/j.enganabound.2009.10.004
J.-S. Chen, L. Wang, H.-Y. Hu, S.-W. Chi, Subdomain radial basis collocation method for heterogeneous media. Int. J. Numer. Meth. Eng. 80(2), 163-190 (2009) https://doi.org/10.1002/nme.2624
N. Mai-Duy, T. Tran-Cong, Compact local integrated-RBF approximations for second-order elliptic differential problems. J. Comput. Phys. 230(12), 4772-4794 (2011) https://doi.org/10.1016/j.jcp.2011.03.002
R.H. Chen, Z.M. Wu, Solving hyperbolic conservation laws using multiquadric quasi interpolation. Numer. Methods Partial Differ. Eq. 22(4), 776-796 (2006) https://doi.org/10.1002/num.20115
Bhatia, G. S., & Arora, G. (2016). Radial basis function methods for solving partial differential equations-a review. Indian Journal of Science and Technology, 9(45), 1-18. https://doi.org/10.17485/ijst/2016/v9i45/105079
Patel, V. G., & Rachchh, N. V. (2020). Meshless method-review on recent developments. Materials today: proceedings, 26, 1598-1603 https://doi.org/10.1016/j.matpr.2020.02.328
A. I. Tolstykh, On using RBF-based differencing formulas for unstructured and mixed structured-unstructured grid calculations, in Proceedings of the 16th IMACS World Congress 228 (Lausanne, 2000), 4606-4624.
A point interpolation meshless method based on radial basis functions,Int.J.Numer. Methods Engrg. 54 (2002), 1623-1648. https://doi.org/10.1002/nme.489
Fornberg, B., & Flyer, N. (2015). A primer on radial basis functions with applications to the geosciences. Society for Industrial and Applied Mathematics https://doi.org/10.1137/1.9781611974041
Milne. (1953). Numerical solution of differential equations. . Wiley, New York.
Lambert, J. D. (1973). Computational methods in ordinary differential equations. Introductory mathematics for scientists and engineers. Wiley, New York.
Singla, R., Singh, G., Ramos, H., & Kanwar, V. (2023). An efficient optimized adaptive step-size hybrid block method for integrating w′′ = f (t, w, w′) directly. Journal of Computational and Applied Mathematics, 420, 114838. https://doi.org/10.1016/j.cam.2022.114838
Omar, Z., & Adeyeye, O. (2016). k-Step block method algorithm for arbitrary order ordinary differential equations. International Journal of Mathematical Analysis, https://doi.org/10.12988/ijma.2016.614
Ngwane, F. F., & Jator, S. N. (2014). L-Stable Block Hybrid Second Derivative Algorithm for Parabolic Partial Differential Equations. American Journal of Computational Mathematics, 04(02), 87-92. https://doi.org/10.4236/ajcm.2014.42008
Olaiya, O. O., Azeez, R. A., & Modebei, M. I. (2021). Efficient Hybrid Block Method For The Numerical Solution Of Second-order Partial Differential Problems via the Method of Lines. Journal of the Nigerian Society of Physical Sciences, 26-37. https://doi.org/10.46481/jnsps.2021.140
Rufai, M. A., & Carpentieri, B. (2023). A variable step-size implementation of a new one-step block method for integrating Burgers' model equation. Alexandria Engineering Journal, 83, 355-366. https://doi.org/10.1016/j.aej.2023.09.078
Mehta, A., Singh, G., & Ramos, H. (2023). Numerical solution of time dependent nonlinear partial differential equations using a novel block method coupled with compact finite difference schemes. Computational and Applied Mathematics, 42(4). https://doi.org/10.1007/s40314-023-02345-3
Gu, Y. T., & Liu, G. R. (2005). A meshfree weak-strong (MWS) form method for time dependent problems. Computational Mechanics, 35(2), 134-145. https://doi.org/10.1007/s00466-004-0610-0
Kaur, Anurag, V. Kanwar, and Higinio Ramos. "An optimized algorithm for numerical solution of coupled Burgers equations." Applied Numerical Mathematics 204 (2024): 352-361. https://doi.org/10.1016/j.apnum.2024.06.019
Shin, S., Sommer, M., Reich, S., & Névir, P. (2010). Evaluation of three spatial discretization schemes with the Galewsky et al. test. Atmospheric Science Letters, 11(3), 223-228. https://doi.org/10.1002/asl.279
Tsai, A. Y. J., Chan, R. P. K., & Wang, S. (2014). Two-derivative Runge-Kutta methods for PDEs using a novel discretization approach. Numerical Algorithms, 65(3), 687-703. https://doi.org/10.1007/s11075-014-9823-2
Le Borne, S., & Leinen, W. (2023). Guidelines for RBF-FD discretization: Numerical experiments on the interplay of a multitude of parameter choices. Journal of scientific computing, 95(1), 8 https://doi.org/10.1007/s10915-023-02123-7
Arora, G., KiranBala, Emadifar, H., & Khademi, M. (2023). A review of radial basis function with applications explored. Journal of the Egyptian Mathematical Society, 31(1), 6. https://doi.org/10.1186/s42787-023-00164-3
Ng, Y. L., Ng, K. C., & Sheu, T. W. H. (2019). A new higher-order RBF-FD scheme with optimal variable shape parameter for partial differential equation. Numerical Heat Transfer, Part B: Fundamentals, 75(5), 289-311. https://doi.org/10.1080/10407790.2019.1627811
Mojarrad, F. N., Veiga, M. H., Hesthaven, J. S., & Öffner, P. (2023). A new variable shape parameter strategy for RBF approximation using neural networks. Computers & Mathematics with Applications, 143, 151-168 https://doi.org/10.1016/j.camwa.2023.05.005
Li, Yang, Dejun Liu, Zhexu Yin, Yun Chen, and Jin Meng. "Adaptive selection strategy of shape parameters for LRBF for solving partial differential equations." Applied Mathematics and Computation 440 (2023): 127645. https://doi.org/10.1016/j.amc.2022.127645
Robert Schaback, Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math. 3 (3) (1995) 251-264 https://doi.org/10.1007/BF02432002
Bengt Fornberg, Cécile Piret, On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere, J. Comput. Phys. 227 (5) (2008) 2758-2780. https://doi.org/10.1016/j.jcp.2007.11.016
Rolland L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res. (1896-1977) 76 (8) (1971) 1905-1915 https://doi.org/10.1029/JB076i008p01905
Richard Franke, Scattered data interpolation: tests of some methods, Math. Comput. 38 (1982) 181-200. https://doi.org/10.1090/S0025-5718-1982-0637296-4
G. Wahba, Spline Models for Observational Data, Society for Industrial and Applied Mathematics, 1990 https://doi.org/10.1137/1.9781611970128
R.E. Carlson, T.A. Foley, The parameter R2 in multi-quadric interpolation, Comput. Math. Appl. 21 (9) (1991) 29-42. https://doi.org/10.1016/0898-1221(91)90123-L
M.L. Stein, Interpolation of Spatial Data: Some Theory for Kriging, Springer, 1999. https://doi.org/10.1007/978-1-4612-1494-6
F. Afiatdoust, M. Esmaeilbeigi, Optimal variable shape parameters using genetic algorithm for radial basis function approximation, Ain Shams Eng. J. 6 (2) (2015) 639-647 https://doi.org/10.1016/j.asej.2014.10.019
J. Biazar, M. Hosami, Selection of an interval for variable shape parameter in approximation by radial basis functions, Adv. Numerical Anal. (2016) https://doi.org/10.1155/2016/1397849
Biazar, J., & Hosami, M. (2017). An interval for the shape parameter in radial basis function approximation. Applied Mathematics and Computation, 315, 131-149. https://doi.org/10.1016/j.amc.2017.07.047
Koupaei, J. A., Firouznia, M., & Hosseini, S. M. M. (2018). Finding a good shape parameter of RBF to solve PDEs based on the particle swarm optimization algorithm. Alexandria engineering journal, 57(4), 3641-3652. https://doi.org/10.1016/j.aej.2017.11.024
Michael Mongillo, Choosing basis functions and shape parameters for radial basis function methods, SIAM Undergrad. Res. Online 4 (190-209) (2011) 2-6. https://doi.org/10.1137/11S010840
Oleg Davydov, Dang Thi Oanh, On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation, Comput. Math. Appl. 62 (5) (2011) 2143-2161. https://doi.org/10.1016/j.camwa.2011.06.037
Bengt Fornberg, Grady Wright, Stable computation of multiquadric interpolants for all values of the shape parameter, Comput. Math. Appl. 48 (5-6) (2004) 853-867. https://doi.org/10.1016/j.camwa.2003.08.010
Grady B. Wright, Bengt Fornberg, Stable computations with flat radial basis functions using vector-valued rational approximations, J. Comput. Phys. 331 (2017) 137-156. https://doi.org/10.1016/j.jcp.2016.11.030
Ramos, Higinio, and Paul Popescu. "How many k-step linear block methods exist and which of them is the most efficient and simplest one?." Applied Mathematics and Computation 316 (2018): 296-309. https://doi.org/10.1016/j.amc.2017.08.036
Ramos, Higinio. "An optimized two-step hybrid block method for solving rst-order initial-value problems in ODEs." Differential Geometry-Dynamical Systems 19 (2017).
Mehta, A., Singh, G., & Ramos, H. (2025). Efficient solution of one-dimensional time-dependent partial differential equations using a multi-step block method combined with compact finite difference schemes. Computational and Applied Mathematics, 44(5), 261. https://doi.org/10.1007/s40314-025-03219-6
Mittal RC, Rohila R (2018) Traveling and shock wave simulations in a viscous Burgers' equation with periodic boundary conditions, Int. J. Appl. Comput. Math. 4 (150). https://doi.org/10.1007/s40819-018-0582-y
Mohanty, R. K., Weizhong Dai, and Fei Han. "Compact operator method of accuracy two in time and four in space for the numerical solution of coupled viscous Burgers' equations." Applied Mathematics and Computation 256 (2015): 381-393. https://doi.org/10.1016/j.amc.2015.01.051
Bak, Soyoon, and Yonghyeon Jeon. "Algorithms for coupled Burgers' equations by sharing characteristic curves within BSLM." Advances in Continuous and Discrete Models 2023, no. 1 (2023): 39. https://doi.org/10.1186/s13662-023-03785-1
A. Ba¸ shan, A numerical treatment of the coupled viscous Burgers' equation in the presence of very large Reynolds number, Phys. A, Stat. Mech. Appl. 545 (2020) 123755. https://doi.org/10.1016/j.physa.2019.123755
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 Results in Nonlinear Analysis
This work is licensed under a Creative Commons Attribution 4.0 International License.
