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

doi:10.1007/s10483-026-3398-8

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

2010 MSC Number:

O302

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

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3398-8

https://www.amm.shu.edu.cn/EN/Y2026/V47/I6/1401

Figures/Tables 15

Fig. 1

Table 1

Values of grid parameters for constructing multi-resolution node distribution"

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

Table 1

Fig. 2

Table 2

Error norms of four numerical solutions for Example 2"

$R e$	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	$1.63 × 10 − 3$	$2.41 × 10 − 4$	$3.45 × 10 − 6$	$6.06 × 10 − 7$	$6.57 × 10 − 8$	$1.12 × 10 − 8$	$4.82 × 10 − 9$	$2.08 × 10 − 9$

Table 2

Fig. 3

Table 3

Error norms of numerical solutions for Example 3 with Re=200"

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	$4.86 × 10 − 5$	$1.17 × 10 − 5$	$4.86 × 10 − 5$	$1.18 × 10 − 5$	$1.52 × 10 − 3$	$5.26 × 10 − 4$	$1.33 × 10 − 6$	$2.66 × 10 − 7$

Table 3

Fig. 4

Fig. 5

Table 4

Error norms of numerical solutions for Example 3 with Re=2 000 and Re=5 000"

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	$1.68 × 10 − 2$	$2.42 × 10 − 3$	$3.56 × 10 − 8$	$1.44 × 10 − 8$	$1.04 × 10 − 7$	$2.63 × 10 − 8$

Table 4

Table 5

Table 6

Fig. 6

Fig. 7

Fig. 8

Table 7

Error norms of the wavelet solution for Example 5"

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	$1.44 × 10 − 6$	$3.46 × 10 − 7$

Table 7

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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