# American Institute of Mathematical Sciences

September  2022, 12(3): 637-658. doi: 10.3934/naco.2021029

## 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 Published  September 2022 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 and Optimization, 2022, 12 (3) : 637-658. doi: 10.3934/naco.2021029
##### References:
 [1] C. Arvanitis, Mesh redistribution strategies and finite element schemes for hyperbolic conservation laws, Journal of Scientific Computing, 34 (2008), 1-25.  doi: 10.1007/s10915-007-9155-7. [2] C. Arvanitis, C. Makridakis and N. Sfakianakis, Entropy conservative schemes and adaptive mesh selection for hyperbolic conservation laws, Journal of Hyperbolic Differential Equations, 7 (2010), 383-404.  doi: 10.1142/S0219891610002177. [3] W. Cao, W. Huang and R. D. Russell, Anr-adaptive finite element method based upon moving mesh PDEs, Journal of Computational Physics, 149 (1999), 221-244.  doi: 10.1006/jcph.1998.6151. [4] T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, MIT Press, 2009. [5] J. M. Coyle, J. E. Flahertya and R. Ludwig, On the stability of mesh equidistribution strategies for time-dependent partial differential equations, Journal of Computational Physics, 62 (1986), 26-39.  doi: 10.1016/0021-9991(86)90098-7. [6] C. De Boor, Good approximation by splines with variable knots. II, In Conference on the numerical solution of differential equations, Springer, (1974), 12–20. [7] E. Dorfi and D. LO'C, Simple adaptive grids for 1-D initial value problems, Journal of Computational Physics, 69 (1987), 175-195. [8] R. K. Dubey and B. Biswas, Suitable diffusion for constructing non-oscillatory entropy stable schemes, Journal of Computational Physics, 372 (2018), 912-930.  doi: 10.1016/j.jcp.2018.04.037. [9] R. K. Dubey and V. Gupta, A mesh refinement algorithm for singularly perturbed boundary and interior layer problems, International Journal of Computational Methods, 17 (2020), 1950024. doi: 10.1142/S0219876219500245. [10] R. Fazio and R. J. LeVeque, Moving-mesh methods for one-dimensional hyperbolic problems using CLAWPACK, Computers & Mathematics with Applications, 45 (2003), 273-298.  doi: 10.1016/S0898-1221(03)80019-6. [11] R. Furzeland, J. Verwer and P. Zegeling, A numerical study of three moving-grid methods for one-dimensional partial differential equations which are based on the method of lines, Journal of Computational Physics, 89 (1990), 349-388.  doi: 10.1016/0021-9991(90)90148-T. [12] M. J. Gander and R. D. Haynes, Domain decomposition approaches for mesh generation via the equidistribution principle, SIAM Journal on Numerical Analysis, 50 (2012), 2111-2135.  doi: 10.1137/110849936. [13] W. Huang, Y. Ren and R. D. Russell, Moving mesh partial differential equations (MMPDES) based on the equidistribution principle, SIAM Journal on Numerical Analysis, 31 (1994), 709-730.  doi: 10.1137/0731038. [14] K. H. Karlsen, K. Brusdal, H. K. Dahle, S. Evje and K. A. Lie, The corrected operator splitting approach applied to a nonlinear advection-diffusion problem, Computer Methods in Applied Mechanics and Engineering, 167 (1998), 239-260.  doi: 10.1016/S0045-7825(98)00122-4. [15] A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations, Journal of Computational Physics, 160 (2000), 241-282.  doi: 10.1006/jcph.2000.6459. [16] C. B. Laney, Computational Gasdynamics, Cambridge University Press, 1998.  doi: 10.1017/CBO9780511605604. [17] S. Li, L. Petzold L and Y. Ren, Stability of moving mesh systems of partial differential equations, SIAM Journal on Scientific Computing, 20 (1998), 719-738.  doi: 10.1137/S1064827596302011. [18] T. Mathew, Domain Decomposition Methods for The Numerical Solution of Partial Differential Equations, Springer Science & Business Media, 61 (2008). doi: 10.1007/978-3-540-77209-5. [19] L. Schwander, D. Ray and J. S. Hesthaven, Controlling oscillations in spectral methods by local artificial viscosity governed by neural networks, Journal of Computational Physics, 431 (2021), 110144. doi: 10.1016/j.jcp.2021.110144. [20] N. Sfakianakis, Finite Difference Schemes on Non-Uniform Meshes for Hyperbolic Conservation Laws, Ph.D thesis, University of Crete, 2009. [21] N. Sfakianakis, Adaptive mesh reconstruction for hyperbolic conservation laws with total variation bound, Mathematics of Computation, 82 (2013), 129-151.  doi: 10.1090/S0025-5718-2012-02615-9. [22] J. M. Stockie, J. A. Mackenzie and R. D. Russell, A moving mesh method for one-dimensional hyperbolic conservation laws, SIAM Journal on Scientific Computing, 22 (2001), 1791-1813.  doi: 10.1137/S1064827599364428. [23] P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM Journal on Numerical Analysis, 21 (1984), 995-1011.  doi: 10.1137/0721062. [24] H. Tang and T. Tang, Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws, SIAM Journal on Numerical Analysis, 41 (2003), 487-515.  doi: 10.1137/S003614290138437X. [25] E. Toro and S. Billett, Centred TVD schemes for hyperbolic conservation laws, IMA Journal of Numerical Analysis, 20 (2000), 47-79.  doi: 10.1093/imanum/20.1.47. [26] A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory, Springer Science & Business Media, 34 (2006). doi: 10.1007/b137868. [27] J. G. Verwer, J. G. Blom, R. Furzeland and P. A. Zegeling, A moving-grid method for one-dimensional PDEs based on the method of lines, CWI; 1988. [28] H. Zhang and P. A. Zegeling, A moving mesh finite difference method for non-monotone solutions of non-equilibrium equations in porous media, Communications in Computational Physics, 22 (2017), 935-964.  doi: 10.4208/cicp.OA-2016-0220. [29] Z. Zhang and T. Tang, An adaptive mesh redistribution algorithm for convection-dominated problems, Communications on Pure & Applied Analysis, 1 (2002), 341-357.  doi: 10.3934/cpaa.2002.1.341.

show all references

##### References:
 [1] C. Arvanitis, Mesh redistribution strategies and finite element schemes for hyperbolic conservation laws, Journal of Scientific Computing, 34 (2008), 1-25.  doi: 10.1007/s10915-007-9155-7. [2] C. Arvanitis, C. Makridakis and N. Sfakianakis, Entropy conservative schemes and adaptive mesh selection for hyperbolic conservation laws, Journal of Hyperbolic Differential Equations, 7 (2010), 383-404.  doi: 10.1142/S0219891610002177. [3] W. Cao, W. Huang and R. D. Russell, Anr-adaptive finite element method based upon moving mesh PDEs, Journal of Computational Physics, 149 (1999), 221-244.  doi: 10.1006/jcph.1998.6151. [4] T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, MIT Press, 2009. [5] J. M. Coyle, J. E. Flahertya and R. Ludwig, On the stability of mesh equidistribution strategies for time-dependent partial differential equations, Journal of Computational Physics, 62 (1986), 26-39.  doi: 10.1016/0021-9991(86)90098-7. [6] C. De Boor, Good approximation by splines with variable knots. II, In Conference on the numerical solution of differential equations, Springer, (1974), 12–20. [7] E. Dorfi and D. LO'C, Simple adaptive grids for 1-D initial value problems, Journal of Computational Physics, 69 (1987), 175-195. [8] R. K. Dubey and B. Biswas, Suitable diffusion for constructing non-oscillatory entropy stable schemes, Journal of Computational Physics, 372 (2018), 912-930.  doi: 10.1016/j.jcp.2018.04.037. [9] R. K. Dubey and V. Gupta, A mesh refinement algorithm for singularly perturbed boundary and interior layer problems, International Journal of Computational Methods, 17 (2020), 1950024. doi: 10.1142/S0219876219500245. [10] R. Fazio and R. J. LeVeque, Moving-mesh methods for one-dimensional hyperbolic problems using CLAWPACK, Computers & Mathematics with Applications, 45 (2003), 273-298.  doi: 10.1016/S0898-1221(03)80019-6. [11] R. Furzeland, J. Verwer and P. Zegeling, A numerical study of three moving-grid methods for one-dimensional partial differential equations which are based on the method of lines, Journal of Computational Physics, 89 (1990), 349-388.  doi: 10.1016/0021-9991(90)90148-T. [12] M. J. Gander and R. D. Haynes, Domain decomposition approaches for mesh generation via the equidistribution principle, SIAM Journal on Numerical Analysis, 50 (2012), 2111-2135.  doi: 10.1137/110849936. [13] W. Huang, Y. Ren and R. D. Russell, Moving mesh partial differential equations (MMPDES) based on the equidistribution principle, SIAM Journal on Numerical Analysis, 31 (1994), 709-730.  doi: 10.1137/0731038. [14] K. H. Karlsen, K. Brusdal, H. K. Dahle, S. Evje and K. A. Lie, The corrected operator splitting approach applied to a nonlinear advection-diffusion problem, Computer Methods in Applied Mechanics and Engineering, 167 (1998), 239-260.  doi: 10.1016/S0045-7825(98)00122-4. [15] A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations, Journal of Computational Physics, 160 (2000), 241-282.  doi: 10.1006/jcph.2000.6459. [16] C. B. Laney, Computational Gasdynamics, Cambridge University Press, 1998.  doi: 10.1017/CBO9780511605604. [17] S. Li, L. Petzold L and Y. Ren, Stability of moving mesh systems of partial differential equations, SIAM Journal on Scientific Computing, 20 (1998), 719-738.  doi: 10.1137/S1064827596302011. [18] T. Mathew, Domain Decomposition Methods for The Numerical Solution of Partial Differential Equations, Springer Science & Business Media, 61 (2008). doi: 10.1007/978-3-540-77209-5. [19] L. Schwander, D. Ray and J. S. Hesthaven, Controlling oscillations in spectral methods by local artificial viscosity governed by neural networks, Journal of Computational Physics, 431 (2021), 110144. doi: 10.1016/j.jcp.2021.110144. [20] N. Sfakianakis, Finite Difference Schemes on Non-Uniform Meshes for Hyperbolic Conservation Laws, Ph.D thesis, University of Crete, 2009. [21] N. Sfakianakis, Adaptive mesh reconstruction for hyperbolic conservation laws with total variation bound, Mathematics of Computation, 82 (2013), 129-151.  doi: 10.1090/S0025-5718-2012-02615-9. [22] J. M. Stockie, J. A. Mackenzie and R. D. Russell, A moving mesh method for one-dimensional hyperbolic conservation laws, SIAM Journal on Scientific Computing, 22 (2001), 1791-1813.  doi: 10.1137/S1064827599364428. [23] P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM Journal on Numerical Analysis, 21 (1984), 995-1011.  doi: 10.1137/0721062. [24] H. Tang and T. Tang, Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws, SIAM Journal on Numerical Analysis, 41 (2003), 487-515.  doi: 10.1137/S003614290138437X. [25] E. Toro and S. Billett, Centred TVD schemes for hyperbolic conservation laws, IMA Journal of Numerical Analysis, 20 (2000), 47-79.  doi: 10.1093/imanum/20.1.47. [26] A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory, Springer Science & Business Media, 34 (2006). doi: 10.1007/b137868. [27] J. G. Verwer, J. G. Blom, R. Furzeland and P. A. Zegeling, A moving-grid method for one-dimensional PDEs based on the method of lines, CWI; 1988. [28] H. Zhang and P. A. Zegeling, A moving mesh finite difference method for non-monotone solutions of non-equilibrium equations in porous media, Communications in Computational Physics, 22 (2017), 935-964.  doi: 10.4208/cicp.OA-2016-0220. [29] Z. Zhang and T. Tang, An adaptive mesh redistribution algorithm for convection-dominated problems, Communications on Pure & Applied Analysis, 1 (2002), 341-357.  doi: 10.3934/cpaa.2002.1.341.
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] 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 [3] Vincent Ducrot, Pascal Frey, Alexandra Claisse. Levelsets and anisotropic mesh adaptation. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 165-183. doi: 10.3934/dcds.2009.23.165 [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] Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221 [6] 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 and Related Models, 2009, 2 (1) : 151-179. doi: 10.3934/krm.2009.2.151 [7] Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6445-6464. doi: 10.3934/dcdsb.2019146 [8] Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems - B, 2009, 11 (3) : 669-690. doi: 10.3934/dcdsb.2009.11.669 [15] 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 [16] 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 [17] Houda Hani, Moez Khenissi. Asymptotic behaviors of solutions for finite difference analogue of the Chipot-Weissler equation. Discrete and 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 and 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 and Continuous Dynamical Systems - B, 2012, 17 (4) : 1155-1174. doi: 10.3934/dcdsb.2012.17.1155

Impact Factor:

## Tools

Article outline

Figures and Tables