# Stability and Resolution Analysis of the Wavelet Collocation Upwind Schemes for Hyperbolic Conservation Laws

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Wavelet Collocation Upwind Schemes

#### 2.1. Wavelet Approximation Theory

#### 2.1.1. Preliminaries

**Ω**is any open subset of

**R**[32]. The space is equipped with the inner product:

**Ω**is denoted by ${\mathbf{C}}^{m}(\mathsf{\Omega})$.

#### 2.1.2. Approximation of Functions on a Finite Domain

**Ω**is a finite domain. We apply the boundary extension technique in our previous study based on the Lagrange interpolation to remove local errors induced by a loss of information outside the domain [33,34]. For all functions in ${\mathbf{L}}^{2}(\mathsf{\Omega})$, the modified wavelet approximation at the resolution level J can be written as

_{n}such that f (x

_{n}) is exactly all the Lagrange interpolations; x

_{k}= k/2

^{J}is the coordinate of the node located inside the domain

**Ω**; ${\phi}_{J,k}(x)=\phi ({2}^{J}x-k)$. In Equation (7) the set {${\tilde{x}}_{i}$, i = 1, 2, …, η} is the collection of selected Lagrange interpolation nodes inside the

**Ω**for the external point x

_{n}and ${\Xi}_{k}={\overline{\Xi}}_{k}\cup \left\{k\right\}$ with the coefficient ${\Pi}_{k}^{k}\left({x}_{k}\right)\equiv 1$. Next, we give two wavelet approximation theorems without proof, which can be proved by using approaches presented in the references [33,35].

**Theorem**

**1.**

_{1,L}and C

_{1,∞}are dependent on the regularity of the derivatives f

^{(n)}(x) but independent of the resolution level J [33].

**Theorem**

**2.**

_{2, L}and C

_{2, ∞}are dependent on the regularity of the derivatives f

^{(n)}(x) but independent of the resolution level J [33].

#### 2.2. Wavelet Collocation Upwind Schemes

**L**

^{2}(

**Ω**). The hyperbolic partial differential Equation (1) can be approximated spatially using the wavelet approximation, resulting in the corresponding ordinary differential equation expressed as follows:

_{T}is the truncation error. The collocation approach can be viewed as a weighted residual method that uses the Kronecker delta function δ

_{J}

_{,l}(x) = δ(2

^{J}x − l) as the weighted function. The collocation discretization is realized by letting the projection of the E

_{T}on a space spanned by the basis {δ

_{J}

_{,l}(x)} equal to zero. Then, the following weighted form can be obtained:

#### 2.3. Asymmetrical Wavelets

_{odd}is greater than that with the N

_{even}when the BM

_{odd}and N

_{odd}are both smaller than the BM

_{even}and N

_{even}by one, respectively. Here, the subscript represents the parity of the N. Some scaling functions are shown in Figure 2 as examples.

## 3. Stability and Resolution Analysis of Wavelet Upwind Schemes

#### 3.1. Advection of a Sine Wave

_{odd}= 1 and BM

_{even}= 2 are taken for the different schemes. The time steps are set small enough to remove the influence of time integration. The numerical errors and orders of accuracy for all schemes are shown in Table 2. It should be noted that errors which are smaller than $5.0\times {10}^{-13}$ are removed from the table since they approach the machine precision of the double type. Theorem 2 shows that the theoretical order of accuracy of the scheme is (N – 1) order. It can be seen that the orders of accuracy are consistent with the expected ones for the schemes where N is smaller than 8. Higher order schemes give the expected orders of accuracy when the number of nodes is small. The N = 4 scheme is unstable even for this smooth wave evolution in our tests. Therefore, the result is absent from Table 2.

#### 3.2. Advection of a Square Wave

#### 3.3. Dissipation and Dispersion Analysis

_{r}= 0 and k

_{i}= kΔx for ideal numerical schemes. When conducting Fourier analysis on Equation (21), we can calculate the following numerical solution:

_{r}is the dissipation coefficient and k

_{i}is the dispersion coefficient. The exact solution for Equation (21) can be computed by

_{r}and k

_{i}. k

_{r}depicts the attenuation of the wave amplitude at t induced by the numerical error, and k

_{i}describes the change in propagation speed of the wave caused by the numerical method. It can be noted that the wave amplitude will be amplified over time if k

_{r}is a negative value. The corresponding scheme shows the instability when solving the advection problems. Therefore, k

_{r}for stable numerical schemes should be non-negative. Here, kΔx is defined as an effective wavenumber and denoted by α. When k

_{i}/α > 1, the numerical wave will run faster than the exact one. Additionally, the numerical wave will fall behind the exact wave for k

_{i}/α < 1. We remark that only the spectral method has the ideal dispersion property, which means that k

_{i}/α = 1.

_{r}decreases with an increase of N. The k

_{r}of the scheme with N ∈ odd is smaller than that with N ∈ even, indicating that the wavelet upwind schemes with N ∈ odd show better dissipation property. To seek the reason for the instability of several schemes, we analyze the k

_{r}for all the above schemes. We find that the negative k

_{r}exists for schemes with N ∈ even. To clarify this fact, the locally enlarged version of Figure 5b is illustrated in Figure 6. It can be observed that k

_{r}are all negative when α < 0.8. This suggests that the schemes with N ∈ even will show a negative diffusion phenomenon when J is large enough, and the error will be amplified over time. This actually induces the instability of the wavelet schemes. We also find that the local extreme value approaches to 0 as N increases for N ∈ even, which means that the negative diffusion process is weakened and the stability of the schemes is improved. This explains how the numerical tests for the schemes with N ∈ even in Section 3.1 and Section 3.2 are still stable. However, for longer time integration and some α near the local extreme value, the results might be divergent. In addition, the reason that the scheme with N = 4, BM = 2 is unstable for the sine wave evolution can also be uncovered based on the dissipation analysis. It can be observed that the scheme with N = 4, BM = 2 has the largest negative dissipation zone of the effective wavenumber α among all the candidates of the wavelet schemes caused by the severest asymmetry, leading to the instability for solving the smooth wave evolution even with far fewer nodes.

_{r}is negative when α is smaller than the specified value. Therefore, the schemes with BM = 3 are unstable. Now, we can achieve a basic instruction that N ∈ odd, BM = 1 schemes have non-negative dissipation coefficients and are stable for hyperbolic conservation laws.

_{i}/ α is approximately equal to 1 depicts the ability of the scheme to trace the wave accurately. We choose the interval of α that satisfies $\left|1-{k}_{i}/\alpha \right|<5\%$ and $\left|1-{k}_{i}/\alpha \right|<2\%$ to measure the maximum α, which reflects the resolution of the schemes directly. The dispersion coefficients against α are plotted in Figure 8. It can be observed that N < 5 schemes have a large dispersion error and a low resolution. The maximum α where k

_{i}/α meets the tolerance relation increases with an increment in N for the specified parity. To clarify the resolution more clearly, we list the maximum α in the tolerance range in Table 4. It can be seen that the maximum α gradually tends to π as N increases, and the schemes with N ∈ even behave better in resolution when N > 6. On the basis of the above analysis, we can find that the wavelets are more applicable to design the high-order schemes, and the scheme with larger N has the higher accuracy and better resolution.

^{−5}, and insert nodes in the adjacent zones near the trouble nodes. Moreover, an integration reconstruction method is designed based on the Lebesgue differentiation theorem to suppress the spurious oscillations. We choose the basic resolution level J

_{0}= 6, the maximum resolution level J

_{max}= 12, and the same adaptive and reconstruction parameters for different schemes. The numerical results obtained by the adaptive wavelet schemes with N ∈ odd and BM = 1 at t = 2 are compared with that of the classic fifth-order finite difference WENO scheme (WENO-5) proposed by Jiang and Shu [38], as illustrated in Figure 10. It can be found that the higher order wavelet schemes can capture the discontinuities without spurious oscillations and distinguish different scale structures accurately with fewer nodes, which also verifies the better resolution of the higher order schemes. For the WENO-5 scheme, a uniform node distribution with N

_{1}= 2048 is required to depict all the details of the solution. The nodes required in the adaptive wavelet upwind schemes are about half of the WENO-5 scheme, showing that the adaptive wavelet schemes with the integration reconstruction can capture discontinuities free from the numerical oscillations and distinguish complex solutions efficiently.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{J}as the base and locate one node in W

_{J}. The BM has the same parity with N. Several candidates are allowed for the positive upwind wavelets with BM > 0.

_{l}= δ

_{0,l}can be easily calculated when l ∈ even. For l ∈ odd, h

_{l}can be computed by

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**Figure 1.**Examples of stencils: (

**a**) Stencil for the asymmetrical wavelet with N = 6 and BM = 2; (

**b**) Stencil for the asymmetrical wavelet with N = 7 and BM = 1.

**Figure 2.**Examples of some asymmetrical scaling functions: (

**a**) Scaling functions with N = 5 and N = 6; (

**b**) Scaling functions with N = 7 and N = 8.

**Figure 3.**Advection of a square wave by the different schemes at t = 8, J = 8: (

**a**) N ∈ odd; (

**b**) N ∈ even.

**Figure 4.**Advection of a square wave by the different schemes at t = 32, J = 8: (

**a**) N ∈ odd; (

**b**) N ∈ even.

**Figure 8.**Dispersion coefficients against α in different wavelet schemes: (

**a**) N ∈ odd; (

**b**) N ∈ even.

**Figure 10.**Numerical results by the schemes with N ∈ odd and BM = 1 at t = 2 (J

_{0}= 6, J

_{max}= 12): (

**a**) N = 5; (

**b**) N = 7; (

**c**) N = 9.

N | 3 | 4 | 5 | 6 | 7 | 7 | 8 | 9 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

BM | 1 | 2 | 1 | 2 | 1 | 3 | 2 | 1 | 3 | 2 |

SF | 0.33 | 0.20 | 0.60 | 0.43 | 0.71 | 0.33 | 0.56 | 0.78 | 0.45 | 0.64 |

Method | N_{1} | l_{∞} Error | l_{∞} Order | l_{2} Error | l_{2} Order |
---|---|---|---|---|---|

N = 3 (2nd order) | 32 | 8.00 × 10^{−2} | — | 8.24 × 10^{−2} | — |

64 | 2.02 × 10^{−2} | 1.99 | 2.05 × 10^{−2} | 2.01 | |

128 | 5.05 × 10^{−3} | 2.00 | 5.08 × 10^{−3} | 2.01 | |

256 | 1.26 × 10^{−3} | 2.00 | 1.27 × 10^{−3} | 2.01 | |

512 | 3.15 × 10^{−4} | 2.00 | 3.16 × 10^{−4} | 2.00 | |

N = 5 (4th order) | 32 | 1.84 × 10^{−4} | — | 1.90 × 10^{−4} | — |

64 | 1.15 × 10^{−5} | 4.01 | 1.16 × 10^{−5} | 4.03 | |

128 | 7.15 × 10^{−7} | 4.00 | 7.21 × 10^{−7} | 4.01 | |

256 | 4.47 × 10^{−8} | 4.00 | 4.49 × 10^{−8} | 4.01 | |

512 | 2.79 × 10^{−9} | 4.00 | 2.80 × 10^{−9} | 4.00 | |

N = 6 (5th order) | 32 | 3.78 × 10^{−5} | — | 3.80 × 10^{−5} | — |

64 | 1.18 × 10^{−6} | 4.99 | 1.19 × 10^{−6} | 5.00 | |

128 | 3.71 × 10^{−8} | 5.00 | 3.71 × 10^{−8} | 5.00 | |

256 | 1.16 × 10^{−9} | 5.00 | 1.16 × 10^{−9} | 5.00 | |

512 | 3.64 × 10^{−11} | 4.99 | 3.64 × 10^{−11} | 4.99 | |

N = 7 (6th order) | 32 | 1.02 × 10^{−6} | — | 1.04 × 10^{−6} | — |

64 | 1.46 × 10^{−8} | 6.12 | 1.48 × 10^{−8} | 6.14 | |

128 | 1.71 × 10^{−10} | 6.42 | 1.73 × 10^{−10} | 6.42 | |

256 | 8.70 × 10^{−13} | 7.61 | 8.75 × 10^{−13} | 7.62 | |

N = 8 (7th order) | 32 | 2.12 × 10^{−7} | — | 2.13 × 10^{−7} | — |

64 | 1.64 × 10^{−9} | 7.01 | 1.64 × 10^{−9} | 7.02 | |

128 | 1.33 × 10^{−11} | 6.95 | 1.33 × 10^{−11} | 6.95 | |

N = 9 (8th order) | 32 | 7.18 × 10^{−9} | — | 7.38 × 10^{−9} | — |

64 | 2.27 × 10^{−11} | 8.31 | 2.30 × 10^{−11} | 8.33 | |

N = 10 (9th order) | 32 | 1.45 × 10^{−9} | — | 1.46 × 10^{−9} | — |

64 | 2.77 × 10^{−12} | 9.03 | 2.77 × 10^{−12} | 9.04 |

Method | N_{1} | l_{∞} Error | l_{∞} Order | l_{2} Error | l_{2} Order |
---|---|---|---|---|---|

N = 7 BM = 1 (6th order) | 32 | 1.02 × 10^{−6} | — | 1.04 × 10^{−6} | — |

64 | 1.46 × 10^{−8} | 6.12 | 1.48 × 10^{−8} | 6.14 | |

128 | 1.71 × 10^{−10} | 6.42 | 1.73 × 10^{−10} | 6.42 | |

256 | 8.70 × 10^{−13} | 7.61 | 8.75 × 10^{−13} | 7.62 | |

N = 7 BM = 3 (6th order) | 32 | 6.94 × 10^{−6} | — | 7.13 × 10^{−6} | — |

64 | 1.10 × 10^{−7} | 5.98 | 1.12 × 10^{−7} | 6.00 | |

128 | 1.78 × 10^{−9} | 5.95 | 1.79 × 10^{−9} | 5.96 | |

256 | 4.20 × 10^{−11} | 5.40 | 3.41 × 10^{−11} | 5.72 | |

512 | 2.01 × 10^{−6} | −15.54 | 1.33 × 10^{−6} | −15.25 | |

N = 9 BM = 1 (8th order) | 32 | 7.18 × 10^{−9} | — | 7.38 × 10^{−9} | — |

64 | 2.27 × 10^{−11} | 8.31 | 2.30 × 10^{−11} | 8.33 | |

N = 9 BM = 3 (8th order) | 32 | 3.79 × 10^{−8} | — | 3.89 × 10^{−8} | — |

64 | 1.53 × 10^{−10} | 7.95 | 1.56 × 10^{−10} | 7.97 | |

128 | 8.73 × 10^{−13} | 7.46 | 8.79 × 10^{−13} | 7.47 |

N | $\left|1-{\mathit{k}}_{\mathit{i}}/\mathit{\alpha}\right|<5\mathit{\%}$ | $\left|1-{\mathit{k}}_{\mathit{i}}/\mathit{\alpha}\right|<2\mathit{\%}$ | N | $\left|1-{\mathit{k}}_{\mathit{i}}/\mathit{\alpha}\right|<5\mathit{\%}$ | $\left|1-{\mathit{k}}_{\mathit{i}}/\mathit{\alpha}\right|<2\mathit{\%}$ |
---|---|---|---|---|---|

3 | 0.397 | 0.247 | 4 | 0.639 | 0.500 |

5 | 1.689 | 1.532 | 6 | 1.249 | 1.012 |

7 | 1.742 | 1.547 | 8 | 2.190 | 1.423 |

9 | 1.847 | 1.652 | 10 | 2.165 | 2.058 |

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**MDPI and ACS Style**

Yang, B.; Wang, J.; Liu, X.; Zhou, Y. Stability and Resolution Analysis of the Wavelet Collocation Upwind Schemes for Hyperbolic Conservation Laws. *Fluids* **2023**, *8*, 65.
https://doi.org/10.3390/fluids8020065

**AMA Style**

Yang B, Wang J, Liu X, Zhou Y. Stability and Resolution Analysis of the Wavelet Collocation Upwind Schemes for Hyperbolic Conservation Laws. *Fluids*. 2023; 8(2):65.
https://doi.org/10.3390/fluids8020065

**Chicago/Turabian Style**

Yang, Bing, Jizeng Wang, Xiaojing Liu, and Youhe Zhou. 2023. "Stability and Resolution Analysis of the Wavelet Collocation Upwind Schemes for Hyperbolic Conservation Laws" *Fluids* 8, no. 2: 65.
https://doi.org/10.3390/fluids8020065