Adaptive wavelet multi-resolution solution for one-dimensional Burgers’ equation at high Reynolds numbers

doi:10.1007/s10483-026-3398-8

摘要/Abstract

Abstract:

The wavelet multi-resolution interpolation Galerkin method (WMIGM) is combined with a mixed explicit-implicit time-stepping scheme to solve the one-dimensional Burgers’ equation at high Reynolds numbers, where the solutions exhibit evolving steep local gradients. In the proposed framework, a dynamic sequence of node distributions with local multi-resolution refinement is adaptively constructed according to the gradient information identified by a wavelet transform. The approximate solution at previous time levels, required in the time-stepping procedure, is represented by the same wavelet expansion used in its original construction, thereby eliminating the need for interpolation between different node distributions. Several representative numerical examples are presented to assess the accuracy, convergence, and robustness of the proposed adaptive wavelet method. The results demonstrate that the proposed approach possesses a higher accuracy and a faster convergence rate than many existing numerical methods, and can accurately capture complex shock dynamics without spurious oscillations, including boundary layer formation from smooth initial profiles and shock merging processes.

Key words: wavelet multi-resolution interpolation, adaptive node distribution, nonlinear convection-diffusion, shock wave

中图分类号:

O302

. [J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(6): 1401-1416.

Jihong ZHENG, Jizeng WANG, Youhe ZHOU, Xiaojing LIU. Adaptive wavelet multi-resolution solution for one-dimensional Burgers’ equation at high Reynolds numbers[J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(6): 1401-1416.

图/表 15

Parameter	Value
Parameter	Some $d m max ⩾ 2 γ ε$	Some $d m max ⩾ ε,$ all $d m max < 2 γ ε$	All $d m max < ε$
$j max n + 1$	$min {j max n + 1, J max}$	$j max n$	$max {j max n − 1, j 0}$
$x ∗ n + 1$	$2 − j max n (k 0 (0) + k 1 (0))$	$2 − j max n (k 0 (1) + k 1 (1))$	$2 − j max n + 1 (k 0 (2) + k 1 (2))$
$R min n + 1$	$2 − j max n + 1 (k 1 (0) − k 0 (0) + γ − 1)$	$2 − j max n ((k 1 (1) − k 0 (1)) / 2 + γ − 1)$	$2 − j max n + 1 (k 1 (2) − k 0 (2) + γ − 1)$
$R max n + 1$	$max {\| k 0 (3) / 2 j 0 − x ∗ n + 1 \|, \| k 1 (3) / 2 j 0 − x ∗ n + 1 \|} + (γ − 1) / 2 j 0 + 1$

$R e$	t	QBFEM (N=50)^[5]		MCBCM (N=50)^[21]		MCBCT4 (N=50)^[23]		Present (N=8)
$R e$	t	$E ∞$	E₂	$E ∞$	E₂	$E ∞$	E₂	$E ∞$	E₂
500	1.68	$4.88 × 10 − 3$	$7.22 × 10 − 4$	$1.72 × 10 − 5$	$3.21 × 10 − 6$	$2.36 × 10 − 7$	$4.43 × 10 − 8$	$1.92 × 10 − 8$	$8.03 × 10 − 9$
	2.10	$4.87 × 10 − 3$	$7.22 × 10 − 4$	$2.23 × 10 − 5$	$4.31 × 10 − 6$	$2.69 × 10 − 7$	$5.25 × 10 − 8$	$2.18 × 10 − 8$	$8.97 × 10 − 9$
	2.66	$4.86 × 10 − 3$	$7.22 × 10 − 4$	$2.92 × 10 − 5$	$5.88 × 10 − 6$	$3.02 × 10 − 7$	$6.18 × 10 − 8$	$2.46 × 10 − 8$	$9.89 × 10 − 9$
	3.30	$4.85 × 10 − 3$	$7.22 × 10 − 4$	$3.73 × 10 − 5$	$7.78 × 10 − 6$	$3.24 × 10 − 7$	$6.99 × 10 − 8$	$2.70 × 10 − 8$	$1.06 × 10 − 8$
1 500	1.68	$1.63 × 10 − 3$	$2.41 × 10 − 4$	$1.48 × 10 − 6$	$2.40 × 10 − 7$	$2.98 × 10 − 8$	$5.35 × 10 − 9$	$2.83 × 10 − 9$	$1.26 × 10 − 9$
	2.10	$1.63 × 10 − 3$	$2.41 × 10 − 4$	$1.97 × 10 − 6$	$3.27 × 10 − 7$	$3.88 × 10 − 8$	$6.75 × 10 − 9$	$3.40 × 10 − 9$	$1.50 × 10 − 9$
	2.66	$1.63 × 10 − 3$	$2.41 × 10 − 4$	$2.65 × 10 − 6$	$4.52 × 10 − 7$	$5.12 × 10 − 8$	$8.76 × 10 − 9$	$4.10 × 10 − 9$	$1.79 × 10 − 9$
	3.30	$1.63 × 10 − 3$	$2.41 × 10 − 4$	$3.45 × 10 − 6$	$6.06 × 10 − 7$	$6.57 × 10 − 8$	$1.12 × 10 − 8$	$4.82 × 10 − 9$	$2.08 × 10 − 9$

t	MCBCM (N=240)^[21]		FECM (N=240)^[24]		RKFIM (N=1200)^[41]		Present (N=65)
t	$E ∞$	E₂	$E ∞$	E₂	$E ∞$	E₂	$E ∞$	E₂
1.70	$9.94 × 10 − 5$	$2.52 × 10 − 5$	$1.00 × 10 − 4$	$2.53 × 10 − 5$	$1.35 × 10 − 3$	$3.84 × 10 − 4$	$1.29 × 10 − 5$	$2.65 × 10 − 6$
2.50	$5.49 × 10 − 5$	$1.51 × 10 − 5$	$5.52 × 10 − 5$	$1.51 × 10 − 5$	$1.55 × 10 − 3$	$4.91 × 10 − 4$	$3.75 × 10 − 6$	$7.65 × 10 − 7$
3.00	$4.14 × 10 − 5$	$1.18 × 10 − 5$	$4.15 × 10 − 5$	$1.19 × 10 − 5$	$1.55 × 10 − 3$	$5.15 × 10 − 4$	$2.24 × 10 − 6$	$4.27 × 10 − 7$
3.50	$4.86 × 10 − 5$	$1.17 × 10 − 5$	$4.86 × 10 − 5$	$1.18 × 10 − 5$	$1.52 × 10 − 3$	$5.26 × 10 − 4$	$1.33 × 10 − 6$	$2.66 × 10 − 7$

t	RKFIM $(R e = 2 000, N = 2 400)$ ^[41]		Present $(N ≈ 260, R e = 2 000)$		Present $(N ≈ 260, R e = 5 000)$
t	$E ∞$	E₂	$E ∞$	E₂	$E ∞$	E₂
1.70	$2.97 × 10 − 2$	$3.59 × 10 − 3$	$1.02 × 10 − 7$	$2.93 × 10 − 8$	$5.60 × 10 − 7$	$1.29 × 10 − 7$
3.00	$1.90 × 10 − 2$	$2.64 × 10 − 3$	$4.30 × 10 − 8$	$1.63 × 10 − 8$	$1.71 × 10 − 7$	$3.73 × 10 − 8$
3.50	$1.68 × 10 − 2$	$2.42 × 10 − 3$	$3.56 × 10 − 8$	$1.44 × 10 − 8$	$1.04 × 10 − 7$	$2.63 × 10 − 8$

t	$ε = 10 − 3,$ $J max = 8$			$ε = 10 − 5,$ $J max = 10$			$ε = 10 − 7,$ $J max = 12$
t	N	$E ∞$	E₂	N	$E ∞$	E₂	N	$E ∞$	E₂
0.1	88	$2.60 × 10 − 4$	$4.37 × 10 − 5$	206	$2.35 × 10 − 6$	$3.53 × 10 − 7$	325	$2.67 × 10 − 7$	$4.59 × 10 − 8$
0.3	80	$2.97 × 10 − 4$	$4.65 × 10 − 5$	190	$2.61 × 10 − 6$	$3.99 × 10 − 7$	319	$4.07 × 10 − 7$	$5.83 × 10 − 8$
0.5	67	$2.50 × 10 − 4$	$4.85 × 10 − 5$	146	$2.75 × 10 − 6$	$3.21 × 10 − 7$	249	$2.69 × 10 − 7$	$4.58 × 10 − 8$
0.7	61	$2.16 × 10 − 3$	$5.40 × 10 − 4$	123	$6.33 × 10 − 6$	$1.23 × 10 − 6$	195	$1.44 × 10 − 6$	$3.46 × 10 − 7$

参考文献 46

[1]	BONKILE, M. P., AWASTHI, A., LAKSHMI, C., MUKUNDAN, V., and ASWIN, V. S. A systematic literature review of Burgers’ equation with recent advances. Pramana, 90(6), 69 (2018)
[2]	RAHMAN, K., HELIL, N., and YIMIN, R. Some new semi-implicit finite difference schemes for numerical solution of Burgers equation. 2010 International Conference on Computer Application and System Modeling, Institute of Electrical and Electronics Engineers, V14-451–V14-455 (2010)
[3]	KUTLUAY, S., BAHADIR, A. R., and ÖZDEŞ A. Numerical solution of one-dimensional Burgers equation: explicit and exact-explicit finite difference methods. Journal of Computational and Applied Mathematics, 103(2), 251–261 (1999)
[4]	WANG, L. L., LIAO, X., and YANG, H. J. A new linearized second-order energy-stable finite element scheme for the nonlinear Benjamin-Bona-Mahony-Burgers equation. Applied Numerical Mathematics, 201, 431–445 (2024)
[5]	RASLAN, K. R. A collocation solution for Burgers equation using quadratic B-spline finite elements. International Journal of Computer Mathematics, 80(7), 931–938 (2003)
[6]	SHENG, Y. and ZHANG, T. The finite volume method for two-dimensional Burgers’ equation. Personal and Ubiquitous Computing, 22(5), 1133–1139 (2018)
[7]	ZHAN, J. M. and LI, Y. S. Generalized finite spectral method for 1D Burgers and KdV equations. Applied Mathematics and Mechanics (English Edition), 27(12), 1635–1643 (2006) https://doi.org/10.1007/s10483-006-1206-z
[8]	KORKMAZ, A. and DAG, İ. Polynomial based differential quadrature method for numerical solution of nonlinear Burgers’ equation. Journal of the Franklin Institute, 348(10), 2863–2875 (2011)
[9]	GAO, Y., LE, L. H., and SHI, B. C. Numerical solution of Burgers’ equation by lattice Boltzmann method. Applied Mathematics and Computation, 219(14), 7685–7692 (2013)
[10]	LIU, X. J., ZHOU, Y. H., ZHANG, L., and WANG, J. Z. Wavelet solutions of Burgers’ equation with high Reynolds numbers. Science China Technological Sciences, 57(7), 1285–1292 (2014)
[11]	LIU, X. J., WANG, J. Z., and ZHOU, Y. H. A space-time fully decoupled wavelet Galerkin method for solving two-dimensional Burgers’ equations. Computers & Mathematics with Applications, 72(12), 2908–2919 (2016)
[12]	LIU, Z. Y., YANG, Y. T., and CAI, Q. D. Neural network as a function approximator and its application in solving differential equations. Applied Mathematics and Mechanics (English Edition), 40(2), 237–248 (2019) https://doi.org/10.1007/s10483-019-2429-8
[13]	MAO, Z. P. and MENG, X. H. Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving partial differential equations with sharp solutions. Applied Mathematics and Mechanics (English Edition), 44(7), 1069–1084 (2023) https://doi.org/10.1007/s10483-023-2994-7
[14]	KADALBAJOO, M. K. and AWASTHI, A. Parameter free hybrid numerical method for solving modified Burgers’ equations on a nonuniform mesh. Asian-European Journal of Mathematics, 10(2), 1750029 (2017)
[15]	HON, Y. C. and MAO, X. Z. An efficient numerical scheme for Burgers’ equation. Applied Mathematics and Computation, 95(1), 37–50 (1998)
[16]	CALDWELL, J., WANLESS, P., and COOK, A. E. Solution of Burgers’ equation for large Reynolds number using finite elements with moving nodes. Applied Mathematical Modelling, 11(3), 211–214 (1987)
[17]	SHYAMAN, V. P., SREELAKSHMI, A., and AWASTHI, A. An adaptive tailored finite point method for the generalized Burgers’ equations. Journal of Computational Science, 62, 101744 (2022)
[18]	CENGIZCI, S. and UĞUR, Ö. A stabilized FEM formulation with discontinuity-capturing for solving Burgers’-type equations at high Reynolds numbers. Applied Mathematics and Computation, 442, 127705 (2023)
[19]	CENGIZCI, S., UĞUR, Ö., and NATESAN, S. A PINN-enhanced SUPG-stabilized hybrid finite element framework with shock-capturing for computing steady convection-dominated flows. Advances in Engineering Software, 216, 104135 (2026)
[20]	JIWARI, R. Local radial basis function-finite difference based algorithms for singularly perturbed Burgers’ model. Mathematics and Computers in Simulation, 198, 106–126 (2022)
[21]	MITTAL, R. C. and JAIN, R. K. Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method. Applied Mathematics and Computation, 218(15), 7839–7855 (2012)
[22]	JIWARI, R. A hybrid numerical scheme for the numerical solution of the Burgers’ equation. Computer Physics Communications, 188, 59–67 (2015)
[23]	SINGH, B. K. and GUPTA, M. A new efficient fourth order collocation scheme for solving Burgers’ equation. Applied Mathematics and Computation, 399, 126011 (2021)
[24]	UÇAR, Y., YAĞMURLU, M., and ÇELİKKAYA, İ. Numerical solution of Burgers’ type equation using finite element collocation method with strang splitting. Mathematical Sciences and Applications E: Notes, 8(1), 29–45 (2020)
[25]	ZHOU, Y. H. Wavelet Numerical Method and Its Applications in Nonlinear Problems, Springer, Singapore (2021)
[26]	WANG, J. Z., LIU, X. J., and ZHOU, Y. H. Application of wavelet methods in computational physics. Annalen der Physik, 536(5), 2300461 (2024)
[27]	MA, X. L., WU, B., ZHANG, J. H., and SHI, X. A new numerical scheme with wavelet-Galerkin followed by spectral deferred correction for solving string vibration problems. Mechanism and Machine Theory, 142, 103623 (2019)
[28]	YU, Q. A homotopy-based wavelet method for extreme large bending analysis of heterogeneous anisotropic plate with variable thickness on orthotropic foundation. Applied Mathematics and Computation, 439, 127641 (2023)
[29]	YU, Q. Wavelet solution for hygrothermomechanical bending of initially defected plate undergoing large deformation on nonlinear elastic foundation. Thin-Walled Structures, 179, 109601 (2022)
[30]	YU, Q. A hierarchical wavelet method for nonlinear bending of materially and geometrically anisotropic thin plate. Communications in Nonlinear Science and Numerical Simulation, 92, 105498 (2021)
[31]	AHMED, S., XU, H., ZHOU, Y., and YU, Q. Modelling convective transport of hybrid nanofluid in a lid driven square cavity with consideration of Brownian diffusion and thermophoresis. International Communications in Heat and Mass Transfer, 137, 106226 (2022)
[32]	AHMED, S., XU, H., WANG, A. Y., and CHEN, Q. B. Highly accurate Coiflet wavelet-homotopy solution of Jeffery-Hamel problem at extreme parameters. International Journal of Wavelets, Multiresolution and Information Processing, 20(5), 2250013 (2022)
[33]	AHMED, S., CHEN, Z. M., XU, H., and ISHAQ, M. Mixed convection flow in a square lid-driven cavity subject to inclined magnetic field with highly accurate wavelet-homotopy solutions. Computers & Mathematics with Applications, 162, 33–51 (2024)
[34]	DIMITRIOU, D. K., NASTOS, C. V., and SARAVANOS, D. A. Multiresolution finite wavelet domain method for efficient modeling of guided waves in composite beams. Wave Motion, 112, 102958 (2022)
[35]	DIMITRIOU, D. K. and SARAVANOS, D. A. Exploring the inherent capacity of the multiresolution finite wavelet domain method to provide convergence indicators in transient dynamic simulations. Computers & Structures, 305, 107517 (2024)
[36]	LIU, X. J., LIU, G. R., WANG, J. Z., and ZHOU, Y. H. A wavelet multiresolution interpolation Galerkin method for targeted local solution enrichment. Computational Mechanics, 64(4), 989–1016 (2019)
[37]	LIU, X. J., ZHOU, Y. H., and WANG, J. Z. Wavelet multiresolution interpolation Galerkin method for nonlinear boundary value problems with localized steep gradients. Applied Mathematics and Mechanics (English Edition), 43(6), 863–882 (2022) https://doi.org/10.1007/s10483-022-2859-5
[38]	LIU, X. J., ZHOU, Y. H., and WANG, J. Z. Highly accurate wavelet solution for bending and free vibration of circular plates over extra-wide ranges of deflections. Journal of Applied Mechanics, 90(3), 031009 (2023)
[39]	DEVILLE, M. O., FISCHER, P. F., and MUND, E. H. High-Order Methods for Incompressible Fluid Flow, Cambridge University Press, Cambridge (2004)
[40]	DONOHO, D. L. Interpolating Wavelet Transforms, Technical Report 408, Stanford University (1992)
[41]	XIE, S. S., HEO, S., KIM, S., WOO, G., and YI, S. Numerical solution of one-dimensional Burgers’ equation using reproducing kernel function. Journal of Computational and Applied Mathematics, 214(2), 417–434 (2008)
[42]	ÖZIŞ, T., AKSAN, E. N., and ÖZDEŞ, A. A finite element approach for solution of Burgers’ equation. Applied Mathematics and Computation, 139(2-3), 417–428 (2003)
[43]	KADALBAJOO, M. K. and AWASTHI, A. A numerical method based on Crank-Nicolson scheme for Burgers’ equation. Applied Mathematics and Computation, 182(2), 1430–1442 (2006)
[44]	JIWARI, R. A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation. Computer Physics Communications, 183(11), 2413–2423 (2012)
[45]	KAKUDA, K. and TOSAKA, N. The generalized boundary element approach to Burgers’ equation. International Journal for Numerical Methods in Engineering, 29(2), 245–261 (1990)
[46]	VAROḠLU, E. and FINN, W. D. L. Space-time finite elements incorporating characteristics for the Burgers’ equation. International Journal for Numerical Methods in Engineering, 16(1), 171–184 (1980)

t	x	Exact	EFDS^[3](N=40)	FEM^[42](N=80)	CNFDS^[43](N=80)	HWCM^[44](N=128)	MHWCM^[22](N=64)	RBFFDS^[20](N=50)	Present (N=32)
0.4	0.25	0.308 89	0.308 91	0.314 29	0.308 81	0.308 87	0.308 89	0.308 90	0.308 89
	0.50	0.569 63	0.569 64	0.576 36	0.569 55	0.569 56	0.569 63	0.569 64	0.569 63
	0.75	0.625 44	0.625 42	0.625 92	0.625 40	0.625 40	0.625 37	0.625 50	0.625 45
1.0	0.25	0.162 56	0.162 57	0.163 91	0.162 54	0.162 55	0.162 58	0.162 57	0.162 57
	0.50	0.291 92	0.291 92	0.294 37	0.291 88	0.291 88	0.291 95	0.291 94	0.291 92
	0.75	0.287 47	0.287 48	0.290 16	0.287 44	0.287 43	0.287 47	0.287 54	0.287 48
3.0	0.25	0.027 20	0.027 20	0.027 43	0.027 20	0.027 21	0.027 20	0.027 19	0.027 20
	0.50	0.040 21	0.040 21	0.040 57	0.040 21	0.040 22	0.040 21	0.040 19	0.040 21
	0.75	0.029 77	0.029 77	0.013 34	0.029 78	0.029 78	0.029 77	0.029 75	0.029 77

t	x	Exact	CNFDS^[43](N=80)	HWCM^[44](N=128)	MHWCM^[22](N=64)	RBFFDS^[20](N=50)	Present (N=32)
5	0.25	0.046 96	0.046 96	0.046 95	0.046 98	0.046 96	0.046 97
	0.50	0.093 92	0.093 93	0.093 91	0.093 96	0.093 92	0.093 92
	0.75	0.140 83	0.140 86	0.140 83	0.140 91	0.140 83	0.140 83
10	0.25	0.024 22	0.024 22	0.024 21	0.024 22	0.024 21	0.024 22
	0.50	0.048 42	0.048 42	0.048 42	0.048 43	0.048 41	0.048 42
	0.75	0.071 13	0.071 12	0.071 14	0.071 18	0.071 12	0.071 14
15	0.25	0.016 31	0.016 31	0.016 31	0.016 31	0.016 31	0.016 31
	0.50	0.032 44	0.032 44	0.032 44	0.032 44	0.032 43	0.032 44
	0.75	0.044 13	0.044 12	0.044 15	0.044 16	0.044 13	0.044 14