Solving nonlinear PDEs using the higher order Haar wavelet method on nonuniform and adaptive grids

    Mart Ratas   Affiliation
    ; Andrus Salupere Affiliation
    ; Jüri Majak   Affiliation


The higher order Haar wavelet method (HOHWM) is used with a nonuniform grid to solve nonlinear partial differential equations numerically. The Burgers’ equation, the Korteweg–de Vries equation, the modified Korteweg–de Vries equation and the sine–Gordon equation are used as model equations. Adaptive as well as nonadaptive nonuniform grids are developed and used to solve the model equations numerically. The numerical results are compared to the known analytical solutions as well as to the numerical solutions obtained by application of the HOHWM on a uniform grid. The proposed methods of using nonuniform grid are shown to significantly increase the accuracy of the HOHWM at the same number of grid points.

Keyword : numerical simulation, Haar wavelet method, higher order wavelet expansion, nonlinear PDEs, nonuniform grid, adaptive grid

How to Cite
Ratas, M., Salupere, A., & Majak, J. (2021). Solving nonlinear PDEs using the higher order Haar wavelet method on nonuniform and adaptive grids. Mathematical Modelling and Analysis, 26(1), 147-169.
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Jan 18, 2021
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M.J. Ablowitz and H. Segur. Solitons and the inverse scattering transform. Society for Industrial and Applied Mathematics, 1981.

E.N. Aksan. A numerical solution of Burgers equation by finite element method constructed on the method of discretization in time. Applied Mathematics and Computation, 170(2):895–904, 2005.

S. Amat and M. Moncayo. Non-uniform multiresolution analysis with supercompact multiwavelets. Journal of Computational and Applied Mathematics, 235(1):334–340, 2010.

I. Aziz and Siraj-ul-Islam. New algorithms for the numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets. Journal of Computational and Applied Mathematics, 239:333–345, 2013.

I. Aziz, Siraj-ul-Islam and F. Khan. A new method based on Haar wavelet for the numerical solution of two-dimensional nonlinear integral equations. Journal of Computational and Applied Mathematics, 272:70–80, 2014.

E. Babolian and A. Shahsavaran. Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets. Journal of Computational and Applied Mathematics, 225(1):87–95, 2009.

C. Basdevant, M. Deville, P. Haldenwang, J.M. Lacroix, J. Ouazzani, R. Peyret, P. Orlandi and A.T. Patera. Spectral and finite difference solutions of the Burgers equation. Comput. & Fluids, 14(1):23–41, 1986.

F. Bashforth and J.C. Adams. An attempt to test the theories of capillary action: by comparing the theoretical and measured forms of drops of fluid. Cambridge University Press, 1883.

E.R. Benton and G.W. Platzman. A table of solutions of the one-dimensional Burgers equation. Quarterly of Applied Mathematics, 30(2):195–212, 1972.

M.J. Berger. Adaptive mesh refinement for hyperbolic partial differential equations. Journal of Computational Physics, 53(3):484–512, 1982.

S. Bertoluzza. An adaptive collocation method based on interpolating wavelets. In W. Dahmen, A.J. Kurdila and P. Oswald(Eds.), Wavelet Analysis and Its Applications, volume 6, pp. 109–135. Elsevier, 1997.

G. Beylkl n and J.M. Keiser. An adaptive pseudo-wavelet approach for solving nonlinear partial differential equations. In W. Dahmen, A.J. Kurdila and P. Oswald(Eds.), Wavelet Analysis and Its Applications, volume 6, pp. 137–197. Elsevier, 1997.

E. Bour. Théorie de la déformation des surfaces. J. Ecole Pol., 1862. Available from Internet:

P. Bowcock, E. Corrigan and C. Zambon. Some aspects of jump-defects in the quantum sine-Gordon model. Journal of High Energy Physics, 2005(08):023, 2005.

C.J. Budd, W. Huang and R.D. Russell. Adaptivity with moving grids. Acta Numerica, 18:111–241, 2009.

F. Bulut, Ö. Oruç and A. Esen. Numerical solutions of fractional system of partial differential equations by Haar wavelets. Computer Modeling in Engineering & Sciences, 108(4):263–284, 2015.

J.M. Burgers. A mathematical model illustrating the theory of turbulence. In R. v. Mises and T. v. Kármán(Eds.), Advances in Applied Mechanics, volume 1, pp. 171–199. Academic Press, 1948.

J.M. Burgers. Statistical problems connected with the solution of a nonlinear partial differential equation. In W.F. Ames(Ed.), Nonlinear Problems of Engineering, pp. 123–137. Academic Press, 1964.

L.M.S. Castro, A.J.M. Ferreira, S. Bertoluzza, R.C. Batra and J.N. Reddy. A wavelet collocation method for the static analysis of sandwich plates using a layerwise theory. Composite Structures, 92(8):1786–1792, 2010.

C. Cattani. Haar wavelet-based technique for sharp jumps classification. Mathematical and Computer Modelling, 39(2):255–278, 2004.

C. Cattani. Haar wavelets based technique in evolution problems. Proceedings of the Estonian Academy of Sciences, 53(1):45–63, 2004.

C. Cattani. On the existence of wavelet symmetries in archaea DNA. Computational and Mathematical Methods in Medicine, 2012, 2012.

C. Cattani and A. Kudreyko. Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind. Applied Mathematics and Computation, 215(12):4164–4171, 2010. .

C. Cattani and Y.Y. Rushchitskii. Cubically nonlinear versus quadratically nonlinear elastic waves: Main wave effects. International Applied Mechanics, 39(12):1361–1399, 2003.

C. Cattani, J.J. Rushchitsky and S.V. Sinchilo. Physical constants for one type of nonlinearly elastic fibrous micro-and nanocomposites with hard and soft nonlinearities. International Applied Mechanics, 41(12):1368–1377, 2005.

C.F. Chen and C.H. Hsiao. Haar wavelet method for solving lumped and distributed-parameter systems. IEE Proceedings - Control Theory and Applications, 144(1):87–94, 1997.

J.D. Cole. On a quasi-linear parabolic equation occurring in aerodynamics. Quarterly of Applied Mathematics, 9(3):225–236, 1951.

C.F. Curtiss and J.O. Hirschfelder. Integration of stiff equations. Proceedings of the National Academy of Sciences of the United States of America, 38(3):235– 243, 1952.

P.G. Drazin and R.S. Johnson. Solitons: an introduction. Cambridge University Press, 1989.

F. Dubeau, S. Elmejdani and R. Ksantini. Non-uniform Haar wavelets. Applied Mathematics and Computation, 159(3):675–693, 2004.

P.R. Eiseman. Adaptive grid generation. Comput. Methods Appl. Mech. Engrg., 64(1-3):321–376, 1987.

Fazal-i-Haq, Siraj-ul-Islam and I. Aziz. Numerical solution of singularly perturbed two-point BVPs using nonuniform Haar wavelets. International Journal for Computational Methods in Engineering Science and Mechanics, 12(4):168– 175, 2011.

F.C. Frank, J.H. van der Merwe and N.F. Mott. One-dimensional dislocations. I. Static theory. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 198(1053):205–216, 1949.

J. Frenkel and T. Kontorova. On the theory of plastic deformation and twinning. Izvestiya Akademii Nauk SSR, Seriya Fizicheskaya, 1:137–149, 1939.

B. Fryxell, K. Olson, P. Ricker, F.X. Timmes, M. Zingale, D.Q. Lamb, P. MacNeice, R. Rosner, J.W. Truran and H. Tufo. FLASH: An adaptive mesh hydrodynamics code for modeling astrophysical thermonuclear flashes. The Astrophysical Journal Supplement Series, 131(1):273–334, 2000. .

C.W. Gear. Numerical initial value problems in ordinary differential equations. Prentice Hall PTR, US, 1971.

M.F. Hamilton and D.T. Blackstock(Eds.). Nonlinear acoustics. Academic Press, San Diego, 1998.

G. Hariharan, K. Kannan and K.R. Sharma. Haar wavelet method for solving Fishers equation. Applied Mathematics and Computation, 211(2):284–292, 2009. ISSN 0096-3003.

H. Hein and L. Feklistova. Computationally efficient delamination detection in composite beams using Haar wavelets. Mechanical Systems and Signal Processing, 25(6):2257–2270, 2011.

A.C. Hindmarsh. ODEPACK, a systematized collection of ODE solvers. Scientific computing, pp. 55–64, 1983.

R. Hirota. Exact solution of the modified Korteweg–de Vries equation for multiple collisions of solitons. Journal of the Physical Society of Japan, 33(5):1456– 1458, 1972.

S.K. Jena and S. Chakraverty. Dynamic behavior of an electromagnetic nanobeam using the Haar wavelet method and the higher-order Haar wavelet method. The European Physical Journal Plus, 134(10):538, 2019.

S.K. Jena, S. Chakraverty and M. Malikan. Implementation of Haar wavelet, higher order Haar wavelet, and differential quadrature methods on buckling response of strain gradient nonlocal beam embedded in an elastic medium. Engineering with Computers, pp. 1–14, 2019.

G. Jin, X. Xie and Z. Liu. The Haar wavelet method for free vibration analysis of functionally graded cylindrical shells based on the shear deformation theory. Composite Structures, 108:435–448, 2014.

J.E. Kim, G.-W. Jang and Y. Y. Kim. Adaptive multiscale wavelet-Galerkin analysis for plane elasticity problems and its applications to multiscale topology design optimization. Internat. J. Solids Structures, 40(23):6473–6496, 2003.

M. Kirs, M. Eerme, D. Bassir and E. Tungel. Application of HOHWM for vibration analysis of nanobeams. In Modern Materials and Manufacturing 2019, volume 799 of Key Engineering Materials, pp. 230–235. Trans Tech Publications Ltd, 5 2019.

A.V. Kravtsov, A.A. Klypin and A.M. Khokhlov. Adaptive refinement tree: a new high-resolution N-body code for cosmological simulations. The Astrophysical Journal Supplement Series, 111(1):73–94, 1997.

M. Kumar and S. Pandit. Wavelet transform and wavelet based numerical methods: an introduction. International Journal of Nonlinear Science, 13(3):325–345, 2012.

S. Kutluay, A.R. Bahadir and A. Özdeş. Numerical solution of onedimensional Burgers equation: explicit and exact-explicit finite difference methods. Journal of Computational and Applied Mathematics, 103(2):251–261, 1999.

P.A. Lagerstrom, J.D. Cole and L. Trilling. Problems in the theory of viscous compressible fluids. Guggenheim Aeronautical Laboratory, unnumbered report, California Institute of Technology, 1949.

G.L. Lamb. Elements of soliton theory. Wiley-Interscience, 1980.

Ü. Lepik. Haar wavelet method for nonlinear integro-differential equations. Applied Mathematics and Computation, 176(1):324–333, 2006.

Ü. Lepik. Application of the Haar wavelet transform to solving integral and differential equations. Proc. Estonian Acad. Sci., 56(1):28–46, 2007.

Ü. Lepik. Numerical solution of evolution equations by the Haar wavelet method. Applied Mathematics and Computation, 185(1):695–704, 2007.

Ü. Lepik. Solving integral and differential equations by the aid of non-uniform Haar wavelets. Applied Mathematics and Computation, 198(1):326–332, 2008.

Ü. Lepik. Haar wavelet method for solving stiff differential equations. Mathematical Modelling and Analysis, 14(4):467–481, 2009.

Ü. Lepik. Solving fractional integral equations by the Haar wavelet method. Applied Mathematics and Computation, 214(2):468–478, 2009.

Ü. Lepik. Solving PDEs with the aid of two-dimensional Haar wavelets. Computers & Mathematics with Applications, 61(7):1873–1879, 2011.

Ü. Lepik and H. Hein. Haar wavelets: with applications. Springer, New York, 2014.

J. Majak, M. Pohlak and M. Eerme. Application of the Haar wavelet-based discretization technique to problems of orthotropic plates and shells. Mechanics of Composite Materials, 45(6):631–642, 2009.

J. Majak, M. Pohlak, M. Eerme and T. Lepikult. Weak formulation based Haar wavelet method for solving differential equations. Applied Mathematics and Computation, 211(2):488–494, 2009. ISSN 0096-3003.

J. Majak, M. Pohlak, M. Eerme and B. Shvartsman. Solving ordinary differential equations with higher order Haar wavelet method. AIP Conference Proceedings, 2116(1):330002, 2019.

J. Majak, M. Pohlak, K. Karjust, M. Eerme, J. Kurnitski and B.S. Shvartsman. New higher order Haar wavelet method: Application to FGM structures. Composite Structures, 201:72–78, 2018. ISSN 0263-8223.

J. Majak, B. Shvartsman, M. Pohlak, K. Karjust, M. Eerme and E. Tungel. Solution of fractional order differential equation by the Haar wavelet method. Numerical convergence analysis for most commonly used approach. AIP Conference Proceedings, 1738(1):480110, 2016.

M B. Mineev and V.V. Shmidt. Radiation from a vortex in a long Josephson junction placed in an alternating electromagnetic field. Sov. Phys. JETP, 52(453):33, 1980.

T. Musha and H. Higuchi. Traffic current fluctuation and the Burgers equation. Japanese Journal of Applied Physics, 17(5):811–816, 1978.

A.C. Newell. Solitons in mathematics and physics. Society for Industrial and Applied Mathematics, 1985.

Ö. Oruç. A non-uniform Haar wavelet method for numerically solving twodimensional convection-dominated equations and two-dimensional near singular elliptic equations. Computers & Mathematics with Applications, 77(7):1799– 1820, 2019.

Ö. Oruç, F. Bulut and A. Esen. A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers equation. Journal of Mathematical Chemistry, 53(7):1592–1607, 2015.

Ö. Oruç, F. Bulut and A. Esen. Numerical solution of the KdV equation by Haar wavelet method. Pramana, 87(6):94, 2016.

Ö. Oruç, F. Bulut and A. Esen. Numerical solutions of regularized long wave equation by Haar wavelet method. Mediterranean Journal of Mathematics, 13(5):3235–3253, 2016.

Ö. Oruç, F. Bulut and A. Esen. A numerical treatment based on Haar wavelets for coupled KdV equation. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 7(2):195–204, 2016.

Ö. Oruç, A. Esen and F. Bulut. A Haar wavelet approximation for twodimensional time fractional reaction–subdiffusion equation. Engineering with Computers, 35(1):75–86, 2019.

L. Petzold. Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations. SIAM journal on scientific and statistical computing, 4(1):136–148, 1983.

M.M. Rai and D.A. Anderson. Application of adaptive grids to fluid-flow problems with asymptotic solutions. AIAA Journal, 20(4):496–502, 1982.

M. Ratas and A. Salupere. Application of higher order Haar wavelet method for solving nonlinear evolution equations. Mathematical Modelling and Analysis, 25(2):271–288, 2020.

J.J. Rushchitsky, C. Cattani and E.V. Terletskaya. Wavelet analysis of the evolution of a solitary wave in a composite material. International Applied Mechanics, 40(3):311–318, 2004.

U. Saeed and M. ur Rehman. Haar wavelet–quasilinearization technique for fractional nonlinear differential equations. Applied Mathematics and Computation, 220:630–648, 2013.

M. Salerno. Discrete model for DNA-promoter dynamics. Physical Review A, 44(8):5292–5297, 1991.

A.C. Scott, F.Y.F. Chu and D.W. McLaughlin. The soliton: A new concept in applied science. Proceedings of the IEEE, 61(10):1443–1483, 1973.

R. Teyssier. Cosmological hydrodynamics with adaptive mesh refinement - a new high resolution code called RAMSES. Astronomy & Astrophysics, 385(1):337– 364, 2002.

H. Triki, T.R. Taha and A.-M. Wazwaz. Solitary wave solutions for a generalized KdV–mKdV equation with variable coefficients. Mathematics and Computers in Simulation, 80(9):1867–1873, 2010.

Siraj ul Islam, I. Aziz and A.S. Al-Fhaid. An improved method based on Haar wavelets for numerical solution of nonlinear integral and integro-differential equations of first and higher orders. Journal of Computational and Applied Mathematics, 260:449–469, 2014.

F. Vazza, G. Brunetti, A. Kritsuk, R. Wagner, C. Gheller and M. Norman. Turbulent motions and shocks waves in galaxy clusters simulated with adaptive mesh refinement. Astronomy & Astrophysics, 504(1):33–43, 2009.

P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright et al. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature methods, 17(3):261–272, 2020.

M. Wang, X. Li and J. Zhang. The -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A, 372(4):417–423, 2008.

S. Wang, Y. Li, Y. Shao, C. Cattani, Y. Zhang and S. Du. Detection of dendritic spines using wavelet packet entropy and fuzzy support vector machine. CNS & Neurological Disorders - Drug Targets, 16(2):116–121, 2017.

X. Xiang, J. Guoyong, L. Wanyou and L. Zhigang. A numerical solution for vibration analysis of composite laminated conical, cylindrical shell and annular plate structures. Composite Structures, 111:20–30, 2014.

X. Xie, G. Jin and Z. Liu. Free vibration analysis of cylindrical shells using the Haar wavelet method. International Journal of Mechanical Sciences, 77:47–56, 2013.