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Fractional diffusion model generalised by the distributed-order operator involving variable diffusion coefficients
Get PDFThe diffusion process plays a crucial role in various fields, such as fluid dynamics, microorganisms, heat conduction and food processing. Since molecular diffusion usually takes place in complex materials and disordered media, there still exist many challenges in describing the diffusion process in the real world. Fractional calculus is a powerful tool for modelling complex physical processes due to its non-local property. This research generalises a fractional diffusion model by using the distributed-order operator in time and the Riesz fractional derivative in space. Moreover, variable diffusion coefficients are introduced to better capture the diffusion complexity. The fractional diffusion model is discretised by the finite element method in space. The approximation of the distributed-order operator is implemented by Simpson’s rule and the L2-1σ formula. A numerical example is provided to verify the effectiveness of the proposed numerical methods. This generalised fractional diffusion model may offer more insights into characterising diffusion behaviours in complex and disordered media.
References
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R. L. Burden, J. D. Faires, and A. M. Burden. Numerical Analysis. Cengage Learning, 2015. url: https://au.cengage.com/c/isbn/9781305253667/
W. Ding, S. Patnaik, S. Sidhardh, and F. Semperlotti. Applications of distributed-order fractional operators: A review. Entropy 23.1 (2021), p. 110. doi: 10.3390/e23010110
G. Gao, A. A. Alikhanov, and Z. Sun. The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J. Sci. Comput. 73.1 (2017), pp. 93–121. doi: 10.1007/s10915-017-0407-x
P. Gouze, Y. Melean, T. Le Borgne, M. Dentz, and J. Carrera. Non-Fickian dispersion in porous media explained by heterogeneous microscale matrix diffusion. Water Resour. Res. 44.11 (2016), pp. 2276–2283. doi: 10.1029/2007WR006690
S. E. Maier, Y. Sun, and R. V. Mulkern. Diffusion imaging of brain tumors. NMR Biomed. 23.7 (2010), pp. 849–864. doi: 10.1002/nbm.1544
M. M. Meerschaert. Fractional calculus, anomalous diffusion, and probability. Fractional Dynamics: Recent Advances. Ed. by J. Klafter, S. C. Lim, and R. Metzler. World Sci., 2011, pp. 265–284. doi: 10.1142/9789814340595_0011
I. Podlubny. Fractional differential equations. New York: Academic Press, 1999. url: https://shop.elsevier.com/books/fractional-differential-equations/podlubny/978-0-12-558840-9
S. Qin, F. Liu, and I. W. Turner. A 2D multi-term time and space fractional Bloch-Torrey model based on bilinear rectangular finite elements. Commun. Nonlin. Sci. Numer. Sim. 56 (2018), pp. 270–286. doi: 10.1016/j.cnsns.2017.08.014
J. N. Reddy. An introduction to the finite element method. Vol. 1221. McGraw-Hill New York, 2004. url: https://www.accessengineeringlibrary.com/content/book/9781259861901
J. P. Roop. Variational solution of the fractional advection dispersion equation. PhD thesis. Clemson University, 2004. url: https://tigerprints.clemson.edu/arv_dissertations/1466/
J. A. Tenreiro Machado, M. F. Silva, R. S. Barbosa, I. S. Jesus, C. M. Reis, M. G. Marcos, and A. F. Galhano. Some applications of fractional calculus in engineering. Math. Prob. Eng. 2010, 639801 (2010). doi: 10.1155/2010/639801
T. Xu, F. Liu, S. Lü, and V. V. Anh. Finite difference/finite element method for two-dimensional time–space fractional Bloch–Torrey equations with variable coefficients on irregular convex domains. In: Comput. Math. App. 80.12 (2020), pp. 3173–3192. doi: 10.1016/j.camwa.2020.11.007
O. C. Zienkiewicz, R. L. Taylor, P. Nithiarasu, and J. Zhu. The finite element method. Vol. 3. McGraw-Hill London, 197
Optimal PML parameters for efficient numerical simulation of waves in an unbounded domain
Get PDFThe perfectly matched layer (PML) is a perfectly non-reflecting layer that simulates the absorption of waves. However, in practice, once the PML is truncated and discretised, the PML is no longer a completely non-reflecting medium. In this article we discuss how to derive optimal PML parameters for the one dimensional acoustic wave equation. Using a multi-block strategy, we present a numerical implementation of the PML that completely eliminates the PML errors. Numerical experiments are presented to verify the analysis.
References
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D. H. Baffet, M. J. Grote, S. Imperiale, and M. Kachanovska. Energy decay and stability of a perfectly matched layer for the wave equation. J. Sci. Comput. 81.3 (2019), pp. 2237–2270. doi: 10.1007/s10915-019-01089-
E. Bécache and M. Kachanovska. Stability and convergence analysis of time-domain perfectly matched layers for the wave equation in waveguides. SIAM J. Numer. Anal. 59.4 (2021), pp. 2004–2039. doi: 10.1137/20M1330543
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J. Diaz and P. Joly. A time domain analysis of PML models in acoustics. Comput. Meth. Appl. Mech. Eng. 195.29 (2006), pp. 3820–3853. doi: 10.1016/j.cma.2005.02.031
K. Duru. The role of numerical boundary procedures in the stability of perfectly matched layers. SIAM J. Sci. Comput. 38.2 (2016), A1171–A1194. doi: 10.1137/140976443
K. Duru and E. M. Dunham. Dynamic earthquake rupture simulations on nonplanar faults embedded in 3D geometrically complex, heterogeneous elastic solids. J. Comput. Phys. 305 (2016), pp. 185–207. doi: 10.1016/j.jcp.2015.10.021
K. Duru, A.-A. Gabriel, and G. Kreiss. On energy stable discontinuous Galerkin spectral element approximations of the perfectly matched layer for the wave equation. Comput. Meth. Appl. Mech. Eng. 350 (2019), pp. 898–937. doi: 10.1016/j.cma.2019.02.036
K. Duru and G. Kreiss. The perfectly matched layer (PML) for hyperbolic wave propagation problems: A review. arXiv, 2201.03733 (2022). doi: 10.48550/ARXIV.2201.03733
T. Lundquist and J. Nordström. The SBP-SAT technique for initial value problems. J. Comput. Phys. 270 (2014), pp. 86–104. doi: 10.1016/j.jcp.2014.03.048
R. Martin and C. Couder-Castaneda. An improved unsplit and convolutional perfectly matched layer absorbing technique for the Navier–Stokes equations using cut-off frequency shift. Comput. Model. Eng. Sci. 63 (2010), pp. 47–77. doi: 10.3970/cmes.2010.063.04
F. Pled and C. Desceliers. Review and recent developments on the perfectly matched layer (PML) method for the numerical modeling and simulation of elastic wave propagation in unbounded domains. Arch. Comput. Meth. Eng. 29 (2021), pp. 471–518. doi: 10.1007/s11831-021-09581-y
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E. Vitanza, R. Grammauta, D. Molteni, and M. Monteforte. A shallow water SPH model with PML boundaries. Ocean Eng. 108 (2015), pp. 315–324. doi: 10.1016/j.oceaneng.2015.07.0
-regular partitions and modular forms with complex multiplication
No full texthttp://dx.doi.org/10.1017/S000497272200139
-adic quotient sets: Linear recurrence sequences
No full texthttp://dx.doi.org/10.1017/S000497271200033
Hecke operators and Drinfeld cusp forms of level t
No full texthttp://dx.doi.org/10.1017/S000497271200033
Canonical decomposition and quiver representations of type Ãn over finite fields
No full texthttp://dx.doi.org/10.1017/S000497271200033
Approximating numbers of the Cantor set by algebraic numbers
No full texthttp://dx.doi.org/10.1017/S000497271200033
The absolute -measure of totally positive algebraic integers
No full texthttp://dx.doi.org/10.1017/S000497271200033
The finite element method for the space fractional magnetohydrodynamic flow and heat transfer on an irregular domain
Get PDFWe consider the magnetohydrodynamic flow and heat transfer of a classical Newtonian fluid in a straight channel with fixed irregular cross section. A spatial fractional operator is introduced to modify the classical Fourier's law of thermal conduction, and we obtain the space fractional coupled model. With the help of the finite element method, the coupled model is solved numerically. Finally, a special numerical example is proposed to verify the stability and efficiency of the presented method.
References
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W. Bu, Y. Tang, Y. Wu, and J. Yang. Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations. J. Comput. Phys. 293 (2015), pp. 264–279. doi: 10.1016/j.jcp.2014.06.031
X. Chi and H. Zhang. Numerical study for the unsteady space fractional magnetohydrodynamic free convective flow and heat transfer with Hall effects. App. Math. Lett. 120, 107312 (2021). doi: 10.1016/j.aml.2021.107312
T. G. Cowling. Magnetohydrodynamics. New York: Interscience, 1957
W. Fan, F. Liu, X. Jiang, and I. Turner. A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain. Frac. Calc. Appl. Anal. 20.2 (2017), pp. 352–383. doi: 10.1515/fca-2017-0019
L. Feng, F Liu, I. Turner, Q. Yang, and P. Zhuang. Unstructured mesh finite difference/finite element method for the 2D time-space Riesz fractional diffusion equation on irregular convex domains. Appl. Math. Model. 59 (2018), pp. 441–463. doi: 10.1016/j.apm.2018.01.044
L. Feng, F. Liu, I. Turner, and L. Zheng. Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette flow of a generalized Oldroyd-B fluid. Frac. Calc. Appl. Anal. 21.4 (2018), pp. 1073–1103. doi: 10.1515/fca-2018-0058
C. Li and A. Chen. Numerical methods for fractional partial differential equations. Int. J. Comp. Math. 95.6–7 (2018), pp. 1048–1099. doi: 10.1080/00207160.2017.1343941
C. Li and F. Zeng. Finite difference methods for fractional differential equations. Int. J. Bifur. Chaos 22.4, 1230014 (2012). doi: 10.1142/S0218127412300145
Y. Liu, X. Chi, H. Xu, and X. Jiang. Fast method and convergence analysis for the magnetohydrodynamic flow and heat transfer of fractional Maxwell fluid. App. Math. Comput. 430, 127255 (2022). doi: 10.1016/j.amc.2022.127255
H. Zhang, F. Liu, and V. Anh. Galerkin finite element approximation of symmetric space-fractional partial differential equations. App. Math. Comput. 217.6 (2010), pp. 2534–2545. doi: 10.1016/j.amc.2010.07.066
H. Zhang, F. Zeng, X. Jiang, and G. E. Karniadakis. Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations. Frac. Calc. Appl. Anal. 25.2 (2022), pp. 453–487. doi: 10.1007/s13540-022-00022-6
M. Zhang, M. Shen, F. Liu, and H. Zhang. A new time and spatial fractional heat conduction model for Maxwell nanofluid in porous medium. Comput. Math. Appl. 78.5 (2019), pp. 1621–1636. doi: 10.1016/j.camwa.2019.01.006
L. Zheng, Y. Liu, and X. Zhang. Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative. Nonlin. Anal.: Real World Appl. 13.2 (2012), pp. 513–523. doi: 10.1016/j.nonrwa.2011.02.01
Numerical solutions to an inverse problem for a non-linear Helmholtz equation
Get PDFIn this work, we develop numerical methods to solve forward and inverse wave problems for a nonlinear Helmholtz equation defined in a spherical shell between two concentric spheres centred at the origin. A spectral method is developed to solve the forward problem while a combination of a finite difference approximation and the least squares method are derived for the inverse problem. Numerical examples are given to verify the method.
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