American Institute of Mathematical Sciences

• Previous Article
On optimal stochastic jumps in multi server queue with impatient customers via stochastic control
• NACO Home
• This Issue
• Next Article
Robust optimum design of tuned mass dampers for high-rise buildings subject to wind-induced vibration
doi: 10.3934/naco.2021029
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

A mesh adaptation algorithm using new monitor and estimator function for discontinuous and layered solution

 1 Department of Mathematics, SRM Institute of Science & Technology, Chennai-603203, India 2 Department of Mathematics, The LNM Institute of Information Technology, Jaipur-302031, India 3 Research Institute & Department of Mathematics, SRM Institute of Science & Technology, Chennai-603203, India

* Corresponding author: Ritesh Kumar Dubey

Received  January 2021 Revised  July 2021 Early access August 2021

In this work a novel mesh adaptation technique is proposed to approximate discontinuous or boundary layer solution of partial differential equations. We introduce new estimator and monitor function to detect solution region containing discontinuity and layered region. Subsequently, this information is utilized along with equi-distribution principle to adapt the mesh locally. Numerical tests for numerous scalar problems are presented. These results clearly demonstrate the robustness of this method and non-oscillatory nature of the computed solutions.

Citation: Prabhat Mishra, Vikas Gupta, Ritesh Kumar Dubey. A mesh adaptation algorithm using new monitor and estimator function for discontinuous and layered solution. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021029
References:

show all references

References:
Plot of Estimator and Monitor function corresponding to step, square and smooth functions
Pseudo-code
Left: Comparing results of Richtmyer two-step Lax Wendroff numerical scheme for linear flux problem over $150$ uniform mesh points and $150$ moving mesh points along with the exact solution and final grid. Right: Mesh point formation at each time level
Left: Comparing results of Richtmyer two-step Lax Wendroff numerical scheme for linear flux problem over $150$ uniform mesh points and $150$ moving mesh points along with the reference solution and final grid. Right: Mesh point formation at each time level
Running-time complexity plot for Example 2
Left: Comparing results of Richtmyer two-step Lax Wendroff numerical scheme for Inviscid Burger's flux problem over $150$ uniform mesh points and $150$ moving mesh points along with the reference solution and final grid. Right: Mesh point formation at each time level
Running-time complexity plot for Example 3
Left: Comparing results of Richtmyer two-step Lax Wendroff numerical scheme for Inviscid Burger's flux problem over $150$ uniform mesh points and $150$ moving mesh points along with the reference solution and final grid. Right: Mesh point formation at each time level
Running-time complexity plot for Example 4
Left: Comparing results of Richtmyer two-step Lax Wendroff numerical scheme for Inviscid Burger's flux problem over $150$ uniform mesh points and $150$ moving mesh points along with the reference solution and final grid. Right: Mesh point formation at each time level
Running-time complexity plot for Example 5
Left: Comparing results of FTCS numerical scheme for Buckley Leverett flux problem over reference solution and $150$ moving mesh points along with the final grid. Right: Mesh point formation at each time level
Running-time complexity plot for Example 6
Left: Comparing results of FTCS numerical scheme for Buckley Leverett flux problem over reference solution and $150$ moving mesh points along with the final grid. Right: Mesh point formation at each time level
Running-time complexity plot for Test problem Example 7
Left: Comparing results of FTCS numerical scheme for Buckley Leverett flux problem using gravitational effect over reference solution and $150$ moving mesh points along with the final grid. Right: Mesh point formation at each time level
Running-time complexity plot for Example 8
Left: Comparing results of FTCS numerical scheme for degenerate parabolic convection-diffusion problem over $150$ uniform mesh points and $150$ moving mesh points along with the final grid. Right: Mesh point formation at each time level
Left: Comparing results of FTCS numerical scheme for non-linear advection diffusion problem over $150$ uniform mesh points and $150$ moving mesh points along with the final grid at $T_f = 1$. Right: Mesh point formation at each time level
Running-time complexity plot for Example 10
Error in discrete $L_{\infty}$ norm, $L_{1}^{loc}$ norm and $L_{2}$ norm for linear advection equation with initial condition (31) at final time $t = 1$
 No. of points $L_{\infty}$ error $L_{1}^{loc}$ error $L_{2}$ error 20 0.56254420515 0.18934971208 0.90499914627 40 0.28834559515 0.10823628178 0.74720768880 80 0.14923240194 0.05220023450 0.53366589869 160 0.07488872339 0.02781070782 0.40332007793 320 0.03729440433 0.01432697423 0.29513421904
 No. of points $L_{\infty}$ error $L_{1}^{loc}$ error $L_{2}$ error 20 0.56254420515 0.18934971208 0.90499914627 40 0.28834559515 0.10823628178 0.74720768880 80 0.14923240194 0.05220023450 0.53366589869 160 0.07488872339 0.02781070782 0.40332007793 320 0.03729440433 0.01432697423 0.29513421904
Average CPU time and number of time-steps of proposed algorithm (ada$150$) as well as numerical solution (uni$150$) with regular grid for each test problem introduced in numerical section
 CPU time and total time steps Test Problem Average CPU time (in sec.) No. of time steps (ada150) (uni150) (ada150) (uni150) Example 2. 2.5796 0.4577 86 5 Example 3. 22.0337 0.9872 730 33 Example 4. 9.9008 0.6591 320 16 Example 5. 2.8051 0.3971 110 5 Example 6. 5.0441 – 200 – Example 7. 5.8861 – 215 – Example 8. 6.1122 – 225 – Example 9. 4.6636 0.5686 180 10 Example 10. 7.9074 1.2978 364 70
 CPU time and total time steps Test Problem Average CPU time (in sec.) No. of time steps (ada150) (uni150) (ada150) (uni150) Example 2. 2.5796 0.4577 86 5 Example 3. 22.0337 0.9872 730 33 Example 4. 9.9008 0.6591 320 16 Example 5. 2.8051 0.3971 110 5 Example 6. 5.0441 – 200 – Example 7. 5.8861 – 215 – Example 8. 6.1122 – 225 – Example 9. 4.6636 0.5686 180 10 Example 10. 7.9074 1.2978 364 70
 [1] Gianluca Frasca-Caccia, Peter E. Hydon. Locally conservative finite difference schemes for the modified KdV equation. Journal of Computational Dynamics, 2019, 6 (2) : 307-323. doi: 10.3934/jcd.2019015 [2] Vincent Ducrot, Pascal Frey, Alexandra Claisse. Levelsets and anisotropic mesh adaptation. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 165-183. doi: 10.3934/dcds.2009.23.165 [3] Claire david@lmm.jussieu.fr David, Pierre Sagaut. Theoretical optimization of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 286-293. doi: 10.3934/proc.2007.2007.286 [4] Emma Hoarau, Claire david@lmm.jussieu.fr David, Pierre Sagaut, Thiên-Hiêp Lê. Lie group study of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 495-505. doi: 10.3934/proc.2007.2007.495 [5] Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6445-6464. doi: 10.3934/dcdsb.2019146 [6] Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221 [7] Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik. On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation. Kinetic & Related Models, 2009, 2 (1) : 151-179. doi: 10.3934/krm.2009.2.151 [8] Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234 [9] Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093 [10] Roumen Anguelov, Jean M.-S. Lubuma, Meir Shillor. Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems. Conference Publications, 2009, 2009 (Special) : 34-43. doi: 10.3934/proc.2009.2009.34 [11] Xiaozhong Yang, Xinlong Liu. Numerical analysis of two new finite difference methods for time-fractional telegraph equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3921-3942. doi: 10.3934/dcdsb.2020269 [12] Allaberen Ashyralyev. Well-posedness of the modified Crank-Nicholson difference schemes in Bochner spaces. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 29-51. doi: 10.3934/dcdsb.2007.7.29 [13] Lih-Ing W. Roeger. Dynamically consistent discrete Lotka-Volterra competition models derived from nonstandard finite-difference schemes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 415-429. doi: 10.3934/dcdsb.2008.9.415 [14] Lili Ju, Wensong Wu, Weidong Zhao. Adaptive finite volume methods for steady convection-diffusion equations with mesh optimization. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 669-690. doi: 10.3934/dcdsb.2009.11.669 [15] Gonzalo Galiano, Julián Velasco. Finite element approximation of a population spatial adaptation model. Mathematical Biosciences & Engineering, 2013, 10 (3) : 637-647. doi: 10.3934/mbe.2013.10.637 [16] Alex Bihlo, James Jackaman, Francis Valiquette. On the development of symmetry-preserving finite element schemes for ordinary differential equations. Journal of Computational Dynamics, 2020, 7 (2) : 339-368. doi: 10.3934/jcd.2020014 [17] Houda Hani, Moez Khenissi. Asymptotic behaviors of solutions for finite difference analogue of the Chipot-Weissler equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1421-1445. doi: 10.3934/dcdss.2016057 [18] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [19] Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4169-4190. doi: 10.3934/dcdsb.2019077 [20] Xiaohai Wan, Zhilin Li. Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1155-1174. doi: 10.3934/dcdsb.2012.17.1155

Impact Factor:

Tools

Article outline

Figures and Tables