次扩散方程L2方法的舍入误差问题及H1范数误差估计

次扩散方程描述了物质在复杂介质中的扩散过程，呈现出具有记忆性的扩散行为。次扩散方程的特性可以很好地被分数阶导数描述，然而求含分数阶导数方程的解析解十分困难，这促使研究者研究分数阶导数的数值近似方法，如多项式插值逼近求解次扩散方程的数值算法。本文主要研究基于二次 Lagrange 插值多项式的 L2 方法，以逼近分数阶 Caputo 导数。致力于解决 L2 方法系数计算中的舍入误差问题和变步长 L2 方法的 𝐻¹范数误差估计。

首先，本文提出一个基于阈值条件和 Taylor 展开（TCTE）的高效处理方法。当阈值条件满足时，直接使用显式公式即可精确计算 L2 方法的系数；当阈值条件不满足时，采用 Taylor 展开的前几项进行近似，即可达到机器误差精度的近似。本文严格给出了相应的舍入误差分析，该技术和分析同样可推广到 L1 和 L2-1𝜎 方法中。通过数值实验对比，本文验证了 TCTE 方法的可行性和高效性。

[1] EINSTEIN A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen [AdP 17, 549 (1905)][J]. Annalen der Physik, 2005, 14(S1 1): 182-193.
[2] VON SMOLUCHOWSKI M. Zur kinetischen theorie der brownschen molekularbewegung undder suspensionen[J]. Annalen der Physik, 1906, 326(14): 756-780.
[3] SAXTON M J. Anomalous diffusion due to obstacles: a Monte Carlo study[J]. Biophysical Journal, 1994, 66(2): 394-401.
[4] SAXTON M J. A biological interpretation of transient anomalous subdiffusion. I. Qualitative model[J]. Biophysical Journal, 2007, 92(4): 1178-1191.
[5] REGNER B M, VUČINIĆ D, DOMNISORU C, et al. Anomalous diffusion of single particles in cytoplasm[J]. Biophysical Journal, 2013, 104(8): 1652-1660.
[6] BRONSTEIN I, ISRAEL Y, KEPTEN E, et al. Transient anomalous diffusion of telomeres in the nucleus of mammalian cells[J]. Physical Review Letters, 2009, 103(1): 018102.
[7] ZSCHOKKE I. Optical spectroscopy of glasses: volume 1[M]. Springer Science & Business Media, 2012.
[8] HUGHES B D. Random walks and random environments[M]. Oxford University Press, 1996.
[9] METZLER R, KLAFTER J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach[J]. Physics Reports, 2000, 339(1): 1-77.
[10] BRUNO L, LEVI V, BRUNSTEIN M, et al. Transition to superdiffusive behavior in intracellular actin-based transport mediated by molecular motors[J]. Physical Review E, 2009, 80(1): 011912.
[11] METZLER R, JEON J H, CHERSTVY A G, et al. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking [J]. Physical Chemistry Chemical Physics, 2014, 16(44): 24128-24164.
[12] METZLER R, TEJEDOR V, JEON J H, et al. Analysis of single particle trajectories: from normal to anomalous diffusion.[J]. Acta Physica Polonica B, 2009, 40(5).
[13] NEUMAN S P, TARTAKOVSKY D M. Perspective on theories of non-Fickian transport in heterogeneous media[J]. Advances in Water Resources, 2009, 32(5): 670-680.
[14] BERKOWITZ B, CORTIS A, DENTZ M, et al. Modeling non-Fickian transport in geologicalformations as a continuous time random walk[J]. Reviews of Geophysics, 2006,44(2).
[15] BARKAI E. Fractional Fokker-Planck equation, solution, and application[J]. Physical Review E, 2001, 63(4): 046118.
[16] MEERSCHAERT M M, SCALAS E. Coupled continuous time random walks in finance[J]. Physica A: Statistical Mechanics and its Applications, 2006, 370(1): 114-118.
[17] YAN H, LAZARIAN A. Cosmic-ray propagation: nonlinear diffusion parallel and perpendicular to mean magnetic field[J]. The Astrophysical Journal, 2008, 673(2): 942.
[18] KLAFTER J, SHLESINGER M F, ZUMOFEN G. Beyond brownian motion[J]. Physics Today, 1996, 49(2): 33-39.
[19] PFISTER G, SCHER H. Dispersive (non-Gaussian) transient transport in disordered solids[J]. Advances in Physics, 1978, 27(5): 747-798.
[20] BERKOWITZ B, SCHER H. Theory of anomalous chemical transport in random fracture networks[J]. Physical Review E, 1998, 57(5): 5858.
[21] KUBO R, TODA M, HASHITSUME N. Statistical physics II: nonequilibrium statistical mechanics: volume 31[M]. Springer Science & Business Media, 2012.
[22] MURALIDHAR R, RAMKRISHNA D, NAKANISHI H, et al. Anomalous diffusion: a dynamic perspective[J]. Physica A: Statistical Mechanics and its Applications, 1990, 167(2): 539-559.
[23] WANG K, DONG L, WU X, et al. Correlation effects, generalized Brownian motion and anomalous diffusion[J]. Physica A: Statistical Mechanics and its Applications, 1994, 203(1): 53-60.
[24] WANG K, TOKUYAMA M. Nonequilibrium statistical description of anomalous diffusion[J]. Physica A: Statistical Mechanics and its Applications, 1999, 265(3-4): 341-351.
[25] WYSS W. The fractional diffusion equation[J]. Journal of Mathematical Physics, 1986, 27(11): 2782-2785.
[26] SCHNEIDER W R, WYSS W. Fractional diffusion and wave equations[J]. Journal of Mathematical Physics, 1989, 30(1): 134-144.
[27] METZLER R, NONNENMACHER T. Fractional diffusion: exact representations of spectral functions[J]. Journal of Physics A: Mathematical and General, 1997, 30(4): 1089.
[28] METZLER R, GLÖCKLE W G, NONNENMACHER T F. Fractional model equation for anomalous diffusion[J]. Physica A: Statistical Mechanics and its Applications, 1994, 211(1): 13-24.
[29] GIONA M, ROMAN H E. Fractional diffusion equation for transport phenomena in random media[J]. Physica A: Statistical Mechanics and its Applications, 1992, 185(1-4): 87-97.
[30] GORENFLO R, MAINARDI F, MORETTI D, et al. Time fractional diffusion: a discrete random walk approach[J]. Nonlinear Dynamics, 2002, 29: 129-143.
[31] PODLUBNY I. Fractional differential equations, mathematics in science and engineering[M]. Academic press New York, 1999.
[32] LAGRANGE J L. Théorie des fonctions analytiques[M]. Imprimerie de la République, 1797.
[33] LUBICH C. Discretized fractional calculus[J]. SIAM Journal on Mathematical Analysis, 1986, 17(3): 704-719.
[34] LUBICH C. Convolution quadrature and discretized operational calculus. I[J]. Numerische Mathematik, 1988, 52(2): 129-145.
[35] LUBICH C. Convolution quadrature revisited[J]. BIT Numerical Mathematics, 2004, 44: 503- 514.
[36] DENG W, HESTHAVEN J S. Local discontinuous Galerkin methods for fractional ordinary differential equations[J]. BIT Numerical Mathematics, 2015, 55(4): 967-985.
[37] MUSTAPHA K. Time-stepping discontinuous Galerkin methods for fractional diffusion problems[J]. Numerische Mathematik, 2015, 130(3): 497-516.
[38] STYNES M, O’RIORDAN E, GRACIA J L. Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation[J]. SIAM Journal on Numerical Analysis, 2017, 55(2): 1057-1079.
[39] KOPTEVA N, MENG X. Error analysis for a fractional-derivative parabolic problem on quasi graded meshes using barrier functions[J]. SIAM Journal on Numerical Analysis, 2020, 58(2): 1217-1238.
[40] KOPTEVA N. Error analysis of an L2-type method on graded meshes for a fractional-order parabolic problem[J]. Mathematics of Computation, 2021, 90(327): 19-40.
[41] LV C, XU C. Error analysis of a high order method for time-fractional diffusion equations[J]. SIAM Journal on Scientific Computing, 2016, 38(5): A2699-A2724.
[42] LIAO H L, MCLEAN W, ZHANG J. A discrete Gronwall inequality with applications to numerical schemes for subdiffusion problems[J]. SIAM Journal on Numerical Analysis, 2019, 57 (1): 218-237.
[43] CHEN H, STYNES M. Error analysis of a second-order method on fitted meshes for a time fractional diffusion problem[J]. Journal of Scientific Computing, 2019, 79: 624-647.
[44] LIAO H L, MCLEAN W, ZHANG J. A second-order scheme with nonuniform time steps for a linear reaction-sudiffusion problem[A]. 2018.
[45] WANG Y M. A high-order compact finite difference method on nonuniform time meshes for variable coefficient reaction–subdiffusion problems with a weak initial singularity[J]. BIT Numerical Mathematics, 2021, 61(3): 1023-1059.
[46] FRANZ S, KOPTEVA N. Pointwise-in-time a posteriori error control for higher-order discretizations of time-fractional parabolic equations[J]. Journal of Computational and Applied Mathematics, 2023, 427: 115122.
[47] JIANG S, ZHANG J, ZHANG Q, et al. Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations[J]. Communications in Computational Physics, 2017, 21(3): 650-678.
[48] LANGLANDS T, HENRY B I. The accuracy and stability of an implicit solution method for the fractional diffusion equation[J]. Journal of Computational Physics, 2005, 205(2): 719-736.
[49] SUN Z Z, WU X. A fully discrete difference scheme for a diffusion-wave system[J]. Applied Numerical Mathematics, 2006, 56(2): 193-209.
[50] LIN Y, XU C. Finite difference/spectral approximations for the time-fractional diffusion equation[J]. Journal of Computational Physics, 2007, 225(2): 1533-1552.
[51] GAO G H, SUN Z Z, ZHANG H W. A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications[J]. Journal of Computational Physics, 2014, 259: 33-50.
[52] ALIKHANOV A A. A new difference scheme for the time fractional diffusion equation[J]. Journal of Computational Physics, 2015, 280: 424-438.
[53] ALIKHANOV A A, HUANG C. A high-order L2 type difference scheme for the time-fractional diffusion equation[J]. Applied Mathematics and Computation, 2021, 411: 126545.
[54] KOPTEVA N. Error analysis of the L1 method on graded and uniform meshes for a fractional derivative problem in two and three dimensions[J]. Mathematics of Computation, 2019, 88 (319): 2135-2155.
[55] LIAO H L, LI D, ZHANG J. Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiffusion equations[J]. SIAM Journal on Numerical Analysis, 2018, 56(2): 1112- 1133.
[56] QUAN C, WU X. H1-norm stability and convergence of an L2-type method on nonuniform meshes for subdiffusion equation[J]. SIAM Journal on Numerical Analysis, 2023, 61(5): 2106-2132.
[57] QUAN C, TANG T, YANG J. How to define dissipation-preserving energy for time-fractional phase-field equations[A]. 2020.
[58] JIN B, LI B, ZHOU Z. Correction of high-order BDF convolution quadrature for fractional evolution equations[J]. SIAM Journal on Scientific Computing, 2017, 39(6): A3129-A3152.
[59] JIN B, LI B, ZHOU Z. Subdiffusion with time-dependent coefficients: improved regularity and second-order time stepping[J]. Numerische Mathematik, 2020, 145(4): 883-913.
[60] JIN B, LAZAROV R, ZHOU Z. Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data[J]. SIAM Journal on Scientific Computing, 2016, 38(1): A146-A170.
[61] MULLER J M, BRISEBARRE N, DE DINECHIN F, et al. Handbook of floating-point arithmetic[M]. Springer, 2018.
[62] GOLDBERG D. What every computer scientist should know about floating-point arithmetic[J]. ACM Computing Surveys (CSUR), 1991, 23(1): 5-48.
[63] QUAN C, WU X. Global-in-time H1-Stability of L2-1σ method on general nonuniform meshes for subdiffusion equation[J]. Journal of Scientific Computing, 2023, 95(2): 594.
[64] QUAN C, WU X, YANG J. Long time H1-stability of fast L2-1σ method on general nonuniform meshes for subdiffusion equations[J]. Journal of Computational and Applied Mathematics, 2024, 440: 115647.
[65] TREFETHEN L N. Spectral methods in MATLAB[M]. SIAM, 2000.
[66] SHEN J, TANG T, WANG L L. Spectral methods: algorithms, analysis and applications: volume 41[M]. Springer Science & Business Media, 2011.