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doi: 10.3934/naco.2021029
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A mesh adaptation algorithm using new monitor and estimator function for discontinuous and layered solution

Prabhat Mishra 1, , Vikas Gupta 2, and  Ritesh Kumar Dubey 3,,

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.

Keywords: Mesh adaptation, Modified equation analysis, finite difference schemes, Evolutionary equations.
Mathematics Subject Classification: 65M06, 65M50.
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:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4] T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, MIT Press, 2009.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[7]

E. Dorfi and D. LO'C, Simple adaptive grids for 1-D initial value problems, Journal of Computational Physics, 69 (1987), 175-195.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16] C. B. Laney, Computational Gasdynamics, Cambridge University Press, 1998.  doi: 10.1017/CBO9780511605604.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

N. Sfakianakis, Finite Difference Schemes on Non-Uniform Meshes for Hyperbolic Conservation Laws, Ph.D thesis, University of Crete, 2009. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory, Springer Science & Business Media, 34 (2006). doi: 10.1007/b137868.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4] T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, MIT Press, 2009.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[7]

E. Dorfi and D. LO'C, Simple adaptive grids for 1-D initial value problems, Journal of Computational Physics, 69 (1987), 175-195.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16] C. B. Laney, Computational Gasdynamics, Cambridge University Press, 1998.  doi: 10.1017/CBO9780511605604.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

N. Sfakianakis, Finite Difference Schemes on Non-Uniform Meshes for Hyperbolic Conservation Laws, Ph.D thesis, University of Crete, 2009. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory, Springer Science & Business Media, 34 (2006). doi: 10.1007/b137868.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Plot of Estimator and Monitor function corresponding to step, square and smooth functions
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Figure 2.  Pseudo-code
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Figure 3.  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
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Figure 4.  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
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Figure 5.  Running-time complexity plot for Example 2
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Figure 6.  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
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Figure 7.  Running-time complexity plot for Example 3
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Figure 8.  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
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Figure 9.  Running-time complexity plot for Example 4
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Figure 10.  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
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Figure 11.  Running-time complexity plot for Example 5
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Figure 12.  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
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Figure 13.  Running-time complexity plot for Example 6
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Figure 14.  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
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Figure 15.  Running-time complexity plot for Test problem Example 7
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Figure 16.  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
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Figure 17.  Running-time complexity plot for Example 8
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Figure 18.  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
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Figure 19.  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
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Figure 20.  Running-time complexity plot for Example 10
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Table 1.  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
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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
Table 2.  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
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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
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