Wavelet-optimized compact finite difference method for convection–diffusion equations
-
Mani Mehra
,
Kuldip Singh Patel
Abstract
In this article, compact finite difference approximations for first and second derivatives on the non-uniform grid are discussed. The construction of diffusion wavelets using compact finite difference approximation is presented. Adaptive grids are obtained for non-smooth functions in one and two dimensions using diffusion wavelets. High-order accurate wavelet-optimized compact finite difference (WOCFD) method is developed to solve convection–diffusion equations in one and two dimensions on an adaptive grid. As an application in option pricing, the solution of Black–Scholes partial differential equation (PDE) for pricing barrier options is obtained using the proposed WOCFD method. Numerical illustrations are presented to explain the nature of adaptive grids for each case.
Funding source: Department of Science and Technology
Award Identifier / Grant number: SERB/F/11946/2018-2019
Acknowledgement
The first author acknowledges the support provided by the Department of Science and Technology, India, under the grant number SERB/F/11946/2018-2019.
-
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
-
Research funding: This study was funded by Department of Science and Technology, India, under grant SERB/F/11946/2018-2019.
-
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] F. Black and M. Scholes, “Pricing of options and corporate liabilities,” J. Polit. Econ., vol. 81, pp. 637–654, 1973, https://doi.org/10.1086/260062.Search in Google Scholar
[2] K. S. Patel and M. Mehra, “A numerical study of Asian option with high-order compact finite difference scheme,” J. Appl. Math. Comput., vol. 57, pp. 467–491, 2018, https://doi.org/10.1007/s12190-017-1115-2.Search in Google Scholar
[3] K. S. Patel and M. Mehra, “High-order compact finite difference scheme for pricing Asian option with moving boundary condition,” Differ. Equ. Dyn. Syst., vol. 27, pp. 39–56, 2019, https://doi.org/10.1007/s12591-017-0372-8.Search in Google Scholar
[4] V. Kumar, “High-order compact finite-difference scheme for singularly-perturbed reaction-diffusion problems on a new mesh of Shishkin type,” J. Optim. Theor. Appl., vol. 143, pp. 123–147, 2009, https://doi.org/10.1007/s10957-009-9547-y.Search in Google Scholar
[5] L. Gamet, F. Ducros, F. Nicoud, and T. Poinsot, “Compact finite difference schemes on non-uniform meshes. Application to direct numerical simulation of compressible flows,” Int. J. Numer. Methods Fluid., vol. 29, pp. 159–191, 1999, https://doi.org/10.1002/(sici)1097-0363(19990130)29:2%3c;159::aid-fld781%3c;3.0.co;2-9.10.1002/(SICI)1097-0363(19990130)29:2<159::AID-FLD781>3.0.CO;2-9Search in Google Scholar
[6] R. K. Shukla and X. Zhong, “Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation,” J. Comput. Phys., vol. 204, pp. 404–429, 2005, https://doi.org/10.1016/j.jcp.2004.10.014.Search in Google Scholar
[7] V. Kumar and B. Srinivasan, “An adaptive mesh strategy for singularly perturbed convection diffusion problems,” Appl. Math. Model., vol. 39, pp. 2081–2091, 2015, https://doi.org/10.1016/j.apm.2014.10.019.Search in Google Scholar
[8] B. Srinivasan, and V. Kumar, “The versatility of an entropy inequality for the robust computation of convection dominated problems,” Procedia Comput. Sci., vol. 108, pp. 887–896, 2017, https://doi.org/10.1016/j.procs.2017.05.099.Search in Google Scholar
[9] M. Mehra, Wavelets Theory and its Applications: A First Course, Singapore, Springer, 2018.10.1007/978-981-13-2595-3Search in Google Scholar
[10] L. Jameson, “On the wavelet-optimized finite difference method,” ICASE CR–191601, 1994.Search in Google Scholar
[11] M. Mehra and V. Kumar, “Fast wavelet-Taylor Galerkin method for linear and non-linear wave problems,” Appl. Math. Comput., vol. 189, pp. 1292–1299, 2007, https://doi.org/10.1016/j.amc.2006.12.013.Search in Google Scholar
[12] B.V. Rathish Kumar and M. Mehra, “Wavelet multilayer Taylor Galerkin schemes for hyperbolic and parabolic problems,” Appl. Math. Comput., vol. 166, pp. 312–323, 2005, https://doi.org/10.1016/j.amc.2004.04.089.Search in Google Scholar
[13] K. Goyal and M. Mehra, “A fast adaptive diffusion wavelet method for Burger’s equation,” Comput. Math. Appl., vol. 68, pp. 568–577, 2014, https://doi.org/10.1016/j.camwa.2014.06.007.Search in Google Scholar
[14] K. Goyal and M. Mehra, “Fast diffusion wavelet method for partial differential equations,” Appl. Math. Model., vol. 40, pp. 5000–5025, 2016, https://doi.org/10.1016/j.apm.2015.10.054.Search in Google Scholar
[15] R. R. Coifman and M Maggioni, “Diffusion wavelets,” Appl. Comput. Harmon. Anal., vol. 21, pp. 53–94, 2006. https://doi.org/10.1016/j.acha.2006.04.004.Search in Google Scholar
[16] M. Mehra and K. S. Patel, “Algorithm 986: A suite of compact finite difference schemes,” ACM Trans. Math Software, vol. 44, 2017, https://doi.org/10.1145/3119905.Search in Google Scholar
[17] K. S. Patel and M. Mehra, “Fourth-order compact scheme for option pricing under the Merton’s and Kou’s jump-diffusion models,” Int. J. Theor. Appl. Finance, vol. 21, 2018, https://doi.org/10.1142/s0219024918500279.Search in Google Scholar
[18] K. S. Patel and M. Mehra, “Fourth-order compact finite difference scheme for American option pricing under regime-switching jump-diffusion models,” Int. J. Appl. Comput. Math., vol. 3, pp. 547–567, 2017, https://doi.org/10.1007/s40819-017-0369-6.Search in Google Scholar
[19] S. K. Lele, “Compact finite difference schemes with spectral-like resolution,” J. Comput. Phys., vol. 103, pp. 16–42, 1992, https://doi.org/10.1016/0021-9991(92)90324-r.Search in Google Scholar
[20] K. S. Patel and M. Mehra, “Fourth-order compact scheme for space fractional advection-diffusion reaction equations with variable coefficients,” J. Comput. Appl. Math., vol. 380, p. 112963, https://doi.org/10.1016/j.cam.2020.112963.Search in Google Scholar
[21] X. Liu, Y. Cao, C. Ma, and L. Shen, “Wavelet-based option pricing: An empirical study,” Eur. J. Oper. Res., vol. 272, pp. 1132–1142, 2019, https://doi.org/10.1016/j.ejor.2018.07.025.Search in Google Scholar
[22] L. Gracia and C. W. Oosterlee, “A highly efficient Shannon wavelet inverse Fourier technique for pricing European options,” SIAM J. Sci. Comput., vol. 38, pp. B118–B143, 2016. https://doi.org/10.1137/15M1014164.Search in Google Scholar
[23] E. Berthe, D. Dang, and L. Gracia, “A Shannon wavelet method for pricing foreign exchange options under the Heston multi-factor CIR model,” Appl. Numer. Math., vol. 136, pp. 1–22, 2019, https://doi.org/10.1016/j.apnum.2018.09.013.Search in Google Scholar
[24] N. Hilber, S. Kehtari, C. Schwab, and C. Winter, “Wavelet finite element method for option pricing in high–dimensional diffusion market models,” Research Report, 2010, 2010-01.Search in Google Scholar
[25] N. Hilber, “Stabilized Wavelet Methods for Option Pricing in High Dimensional Stochastic Volatility Models,” PhD thesis, Zrich, ETH, 2009.Search in Google Scholar
[26] R. M. X. Rometsch, “A Wavelet Tour of Option Pricing,” PhD thesis, Germany, University of Ulm, Baden-Wrttemberg, 2010.Search in Google Scholar
[27] B. C. D. Wiart and M. A. H. Dempster, “Wavelet optimized valuation of financial derivatives,” Int. J. Theor. Appl. Finance, vol. 14, pp. 1113–1137, 2011. https://doi.org/10.1142/S021902491100667X.Search in Google Scholar
[28] L. Gracia and C. W. Oosterlee, “Robust pricing of European options with wavelets and the characteristic function,” SIAM J. Sci. Comput., vol. 35, pp. B1055–B1084, 2013.10.1137/130907288Search in Google Scholar
[29] D. Dang and L. Gracia, “A dimension reduction Shannon-wavelet based method for option pricing,” J. Sci. Comput., vol. 75, pp. 733–761, 2018, https://doi.org/10.1007/s10915-017-0556-y.Search in Google Scholar
[30] A. Biswas, A. Goswami, and L. Overbeck, “Option pricing in a regime switching stochastic volatility model,” Stat. Probab. Lett., vol. 138, pp. 116–126, 2018, https://doi.org/10.1016/j.spl.2018.02.056.Search in Google Scholar
[31] M. K. Das, A. Goswami, and T. S. Patankar, “Pricing derivatives in a regime switching market with time inhomogenous volatility,” Stoch. Anal. Appl., vol. 36, pp. 700–725, 2018. https://doi.org/10.1080/07362994.2018.1448996.Search in Google Scholar
[32] A. Goswami, J. Patel, and P. Shevgaonkar, “A system of non-local parabolic PDE and application to option pricing,” Stoch. Anal. Appl., vol. 34, pp. 893–905, 2016, https://doi.org/10.1080/07362994.2016.1189340.Search in Google Scholar
[33] S. T. Lee and H. W. Sun, “Fourth order compact scheme with local mesh refinement for option pricing in jump-diffusion model,” Numer. Methods Part. Differ. Equ., vol. 28, pp. 1079–1098, 2011, https://doi.org/10.1002/num.20677.Search in Google Scholar
[34] H. O. Kreiss, V. Thomee, and O. Widlund, “Smoothing of initial data and rates of convergence for parabolic difference equations,” Commun. Pure Appl. Math., vol. 23, pp. 241–259, 1970, https://doi.org/10.1002/cpa.3160230210.Search in Google Scholar
[35] D. Y. Tangman, A. Gopaul, and M. Bhuruth, “Numerical pricing of options using high-order compact finite difference schemes,” J. Comput. Appl. Math., vol. 218, pp. 270–280, 2008, https://doi.org/10.1016/j.cam.2007.01.035.Search in Google Scholar
[36] J. C. Strikewerda, Finite Difference Schemes and Partial Differential Equations, Philadelphia, SIAM, 2004.10.1137/1.9780898717938Search in Google Scholar
[37] I. Daubechies, Ten Lectures on Wavelets, Philadelphia, SIAM, 1992.10.1137/1.9781611970104Search in Google Scholar
[38] M. J. NG Stynes, E. O’riordon, and M. Stynes, “Numerical methods for time dependent convection–diffusion equations,” J. Comput. Appl. Math., vol. 21, pp. 289–310, 1988, https://doi.org/10.1016/0377-0427(88)90315-9.Search in Google Scholar
[39] S. Zhang and W. Wang, “A stencil of the finite-difference method for the 2D convection diffusion equation and its new iterative scheme,” Int. J. Comput. Math., vol. 87, pp. 2588–2600, 2010, https://doi.org/10.1080/00207160802691637.Search in Google Scholar
[40] P. Brandimarte, Numerical Methods in Finance and Economics: A MATLAB Based Introduction, Hoboken, New Jersey, John Wiley & Sons, 2006.10.1002/0470080493Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Numerical analysis of geometrical nonlinear aeroelasticity with CFD/CSD method
- Application of moving least squares algorithm for solving systems of Volterra integral equations
- Recurrence relations for a family of iterations assuming Hölder continuous second order Fréchet derivative
- Stochastic models with multiplicative noise for economic inequality and mobility
- Fast interaction of Cu2+ with S2O3 2− in aqueous solution
- Rotorcraft fuselage/main rotor coupling dynamics modelling and analysis
- Modeling and fault identification of the gear tooth surface wear failure system
- Wavelet-optimized compact finite difference method for convection–diffusion equations
- A methodology for dynamic behavior analysis of the slider-crank mechanism considering clearance joint
- Dynamics of a stochastic susceptible-infective-recovered (SIRS) epidemic model with vaccination and nonlinear incidence under regime switching and Lévy jumps
- The twin properties of rogue waves and homoclinic solutions for some nonlinear wave equations
- Distributed consensus of multi-agent systems with increased convergence rate
- Studies on population balance equation involving aggregation and growth terms via symmetries
- Bifurcation, routes to chaos, and synchronized chaos of electromagnetic valve train in camless engines
- Feedback control of a nonlinear aeroelastic system with non-semi-simple eigenvalues at the critical point of Hopf bifurcation
- Nonlinear dynamics of a RLC series circuit modeled by a generalized Van der Pol oscillator
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Numerical analysis of geometrical nonlinear aeroelasticity with CFD/CSD method
- Application of moving least squares algorithm for solving systems of Volterra integral equations
- Recurrence relations for a family of iterations assuming Hölder continuous second order Fréchet derivative
- Stochastic models with multiplicative noise for economic inequality and mobility
- Fast interaction of Cu2+ with S2O3 2− in aqueous solution
- Rotorcraft fuselage/main rotor coupling dynamics modelling and analysis
- Modeling and fault identification of the gear tooth surface wear failure system
- Wavelet-optimized compact finite difference method for convection–diffusion equations
- A methodology for dynamic behavior analysis of the slider-crank mechanism considering clearance joint
- Dynamics of a stochastic susceptible-infective-recovered (SIRS) epidemic model with vaccination and nonlinear incidence under regime switching and Lévy jumps
- The twin properties of rogue waves and homoclinic solutions for some nonlinear wave equations
- Distributed consensus of multi-agent systems with increased convergence rate
- Studies on population balance equation involving aggregation and growth terms via symmetries
- Bifurcation, routes to chaos, and synchronized chaos of electromagnetic valve train in camless engines
- Feedback control of a nonlinear aeroelastic system with non-semi-simple eigenvalues at the critical point of Hopf bifurcation
- Nonlinear dynamics of a RLC series circuit modeled by a generalized Van der Pol oscillator
