# A Sequential, Implicit, Wavelet-Based Solver for Multi-Scale Time-Dependent Partial Differential Equations

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## Abstract

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**MSC**65T60; 65M20

## 1. Introduction

## 2. Wavelet Decomposition

**Figure 1.**Top: The wavelet pair corresponding to the spline function ${B}_{3}$, its second derivative and dual re-centered at $x=0$. Bottom: The same for the corresponding wavelet. (

**a**) $\varphi \left(x\right)$; (

**b**) $\frac{{d}^{2}}{d{x}^{2}}\phantom{\rule{0.166667em}{0ex}}\varphi \left(x\right)$; (

**c**) $\tilde{\varphi}\left(x\right)$; (

**d**) $\psi \left(x\right)$; (

**e**) $\frac{{d}^{2}}{d{x}^{2}}\psi \left(x\right)$; (

**f**) $\tilde{\psi}\left(x\right)$.

## 3. Multi-Scale Method

## 4. Implementation

**Example 4.1**

## 5. Error Sources

- The vectors ${\mathbf{b}}_{1}$ not being included at all. In order to keep the error below a specified tolerance, we would need to implement an adaptive procedure. This would involve calculating the values of the coefficients ${b}_{j,k}$ in ${\mathbf{b}}_{1}$ that are adjacent to the domain Λ. The domain Λ would be expanded if these values were found to have exceeded the specified threshold. Other coefficients ${b}_{j,k}$ in ${\mathbf{b}}_{1}$, further from Λ, would also be calculated. If these were found to be above the specified threshold, then the large-scale system would be given a finer resolution.
- The matrices ${C}_{2}$ and ${B}_{3}$ from (5.6) being omitted from (5.9). This source of error is reduced by widening the vectors ${\mathbf{a}}_{\Lambda}$ at the expense of ${\mathbf{a}}_{1}$ (i.e., including extra large-scale coefficients in the calculation of the small-scale system). We call these “extra” terms, and they are explained in detail in Section 8. These terms reduce the error because the matrices A to D will be diagonally dominant (due to the fact that φ and ψ are well centered). Consider the block decomposition of$$C=\left[\begin{array}{cc}{C}_{1}& {C}_{2}\\ {C}_{3}& {C}_{\Lambda}\end{array}\right]$$The best way to control this source of error is an adaptive scheme that checks the largest values of ${C}_{2}{\mathbf{b}}_{\Lambda}$ and ${B}_{3}{\mathbf{a}}_{1}$. The number of “extra” terms would be increased if either calculation was above a given tolerance.
- The last type of error is from the last vector term in (5.9). This source of error is best managed via the “extra” terms, as well (i.e., by increasing the number of components in ${\mathbf{a}}_{\Lambda}$). Including more “extra” terms will result in ${D}_{2}$ having fewer elements, as well as smaller elements (as with ${C}_{2}$ and ${B}_{3}$ above). One other thing to notice is that ${\mathbf{a}}_{\Lambda}^{{C}_{r}}$ is a corrector term, and so, in general, is of smaller magnitude than ${\mathbf{a}}_{\Lambda}^{n}$. We have$$-\frac{\phantom{\rule{0.166667em}{0ex}}h\phantom{\rule{0.166667em}{0ex}}}{2}\phantom{\rule{0.166667em}{0ex}}{D}_{2}\phantom{\rule{0.166667em}{0ex}}{\mathbf{a}}_{\Lambda}^{{C}_{r}}=-\frac{{h}^{2}}{4}\phantom{\rule{0.166667em}{0ex}}{D}_{2}\left[{C}_{\Lambda}({\mathbf{b}}_{\Lambda}^{n}+{\mathbf{b}}_{\Lambda}^{n+1})+{D}_{\Lambda}{\mathbf{a}}_{\Lambda}^{{C}_{r}}\right]$$

## 6. A Test Problem

**Figure 3.**Results for Burgers’ equation with ${V}_{-6}$ resolution and time step $h=0.01$. (

**a**) $t=0.0$; (

**b**) $t=0.5$; (

**c**) $t=1.0$; (

**d**) $t=1.5$; (

**e**) $t=2.0$; (

**f**) $t=2.5$.

## 7. Calculation

#### 7.1. Time-Discretization

#### 7.2. Physical Decomposition

#### 7.3. Results

**Figure 4.**Results for Burgers’ equation using ${V}_{-8}$, and time step size $h=0.002$, (

**a**) at $t=1.0$ and (

**b**) at $t=2.0$. Next, the difference between the ${V}_{-8}$ results and the ${V}_{-6}$ results, (

**c**) at $t=1.0$ and (

**d**) at $t=2.0$.

## 8. Further Improvements

#### 8.1. Non-Linear Terms

#### 8.2. “Extra” Terms

**Figure 5.**A simple diagram showing Ω, Λ, ${\Lambda}^{\prime}$ and the location of the “extra” terms.

## 9. Results

**Figure 6.**Results for Burgers’ equation using the components from Section 8. These use ${V}_{-4}$ with localized ${V}_{-6}$ and three “extra” coefficients ${a}_{-3,k}$ around $\Lambda =(3.75,6.25)$. It also uses ${\left({J}_{NL}\right)}_{i,i}$ values of 0, 0, 0, 0, $\frac{1}{2}$, 1, 1, etc. The time step size is $h=0.01$. (

**a**) $t=0.5$; (

**b**) $t=1.0$; (

**c**) $t=1.5$; (

**d**) $t=2.0$; (

**e**) $t=2.5$; (

**f**) $t=3.0$.

**Figure 8.**Error from results using ${V}_{-5}$ with localized ${V}_{-6}$. The error is the difference between the multi-scale results and the ${V}_{-6}$ control results, all calculated at $t=2.0$. The multi-scale results differ in their Λ sub-domains. All use 4 “extra” terms per boundary of Λ. (

**a**) $\Lambda =(3.75,6.25)$; (

**b**) $\Lambda =(3.50,6.50)$; (

**c**) $\Lambda =(2.50,7.50)$; (

**d**) $\Lambda =(1.50,8.50)$.

**Figure 9.**Error from multi-scale systems, compared with system size and computational time. The system size is the total number of coefficients ${a}_{j,k}$ and ${b}_{j,k}$ used in the systems (also seen in Table 1). The computational time is the average time taken to calculate a single time step, in seconds (all using the same computer). The vertical axis is the ${log}_{10}$ of the error, the difference between the multi-scale results and the fine resolution control results on (1.5, 8.5). (

**a**) ${log}_{10}$ of the ${L}^{2}$ error; (

**b**) ${log}_{10}$ of the ${L}^{1}$ error; (

**c**) ${log}_{10}$ of the ${L}^{\infty}$ error.

**Table 1.**Burgers’ equation error for multi-scale systems. By “Time” we mean the average computing time (in seconds) per time step. Note that the errors are only calculated in the region $[1.5,8.5]$.

Λ | “extra” | Terms | Time | ${L}_{2}$ Error | ${L}_{1}$ Error | ${L}_{\infty}$ Error |
---|---|---|---|---|---|---|

$[3.75,6.25]$ | 4 | 128 | 0.0904 | $2.4\times {10}^{-5}$ | $4.6\times {10}^{-5}$ | $3.7\times {10}^{-5}$ |

$[3.50,6.50]$ | 5 | 138 | 0.0998 | $1.6\times {10}^{-6}$ | $2.8\times {10}^{-6}$ | $2.1\times {10}^{-6}$ |

$[3.00,7.00]$ | 6 | 156 | 0.1241 | $1.0\times {10}^{-7}$ | $2.1\times {10}^{-7}$ | $1.2\times {10}^{-7}$ |

$[2.50,7.50]$ | 7 | 174 | 0.1560 | $5.8\times {10}^{-9}$ | $1.4\times {10}^{-8}$ | $6.2\times {10}^{-9}$ |

$[2.00,8.00]$ | 7 | 190 | 0.1977 | $3.7\times {10}^{-9}$ | $8.6\times {10}^{-9}$ | $4.1\times {10}^{-9}$ |

$[1.50,8.50]$ | 7 | 206 | 0.2541 | $2.1\times {10}^{-9}$ | $2.0\times {10}^{-9}$ | $4.6\times {10}^{-9}$ |

${V}_{-6}$ control results | 0.9888 |

## 10. A Further Test

**Figure 10.**The spline function ${B}_{5}=\varphi $, our scaling function. Next, $\frac{{d}^{4}}{d{x}^{4}}\phantom{\rule{0.166667em}{0ex}}\varphi $ and our wavelet, ψ. (

**a**) $\varphi \left(x\right)$; (

**b**) $\frac{{d}^{4}}{d{x}^{4}}\phantom{\rule{0.166667em}{0ex}}\varphi \left(x\right)$; (

**c**) $\psi \left(x\right)$.

#### 10.1. The Rossby Wave Problem

**Figure 11.**(Top) The flux (

**a**), steady state for P (

**b**). (Bottom) the progression of Z at $t=50,75,100$ in the x-y plane, from left to right. These use $\delta =0.2$, $\beta =1$ and $\nu =\u03f5=0$. Notice that Z increases in amplitude and decreases in wavelength as time increases. (

**a**) The Flux; (

**b**) P; (

**c**) Z, $t=50$; (

**d**) Z, $t=75$; (

**e**) Z, $t=100$.

#### 10.2. Linear Results

**Figure 12.**The results from a linear ${V}_{-6}$ system. Notice that $F\left(t\right)$ (the x averaged momentum flux) remains stable until $t=200$. These results use a “switch-on” function for the boundary condition values equal to $min\left\{\frac{t}{5},1\right\}$, $\beta =1$, $\delta =0.2$, and no viscosity. (

**a**) $F\left(t\right)$; (

**b**) P, $t=200$; (

**c**) Z, $t=200$.

**Figure 13.**The results from a linear ${V}_{-5}$ system. Notice that $F\left(t\right)$ remains stable until $t=100$, where Z is following the same pattern seen in Figure 12c. These results are based around the same problem as those in Figure 12, just with a different resolution. (

**a**) $F\left(t\right)$; (

**b**) P, $t=100$; (

**c**) Z, $t=100$.

**Figure 14.**Results from a ${V}_{-4}$ with localized ${V}_{-6}$ double system. The small-scale is over $\Lambda =(3.0,6.5)$, and 3 “extra” terms are used over both interfaces of Λ. There is a little instability in the flux around $t=200$, but this arrangement is very close to the full ${V}_{6}$ control system, found in Figure 12. (

**a**) $F\left(t\right)$; (

**b**) P, $t=200$; (

**c**) Z, $t=200$.

#### 10.3. Non-Linear Results

**Figure 15.**Plots for the $\u03f5=0.01$ problem using ${V}_{-4}$. (Top) Flux; (Middle) P; (Bottom) Z. (

**a**) P, $t=10$; (

**b**) P, $t=30$; (

**c**) P, $t=50$; (

**d**) Z, $t=10$; (

**e**) Z, $t=30$; (

**f**) Z, $t=50$.

**Figure 16.**Plots for ${V}_{-3}:{V}_{-4}$ multi-scale models of the $\u03f5=0.01$ Rossby wave problem. All use $\Lambda =(1.5,6.5)$. The flux is plotted against that for the ${V}_{-4}$ control results (the lighter, thicker line). The results using the larger $\Lambda =(3.5,9.0)$ and $\Lambda =(3.5,9.5)$ are omitted, as they are visually identical to the full ${V}_{-4}$ control results from Figure 15. (

**a**) $F\left(t\right)$; (

**b**) P at $t=50$; (

**c**) Z at $t=50$.

#### 10.4. Convergence

**Figure 17.**The error at $t=200$ of the function Z from the linear problem. The vertical axis is the ${log}_{10}$ of the error (the difference between the multi-scale system results and the ${V}_{-6}$ control results). The horizontal axis is the total number of ${a}_{j,k}$ and ${b}_{j,k}$ coefficients used for the y directional decomposition. (

**a**) ${log}_{10}$ of the ${L}^{2}$ error for ${V}_{-4}$ with ${V}_{-6}$ on Λ; (

**b**) ${log}_{10}$ of the ${L}^{2}$ error for ${V}_{-5}$ with ${V}_{-6}$ on Λ; (

**c**) ${log}_{10}$ of the ${L}^{1}$ error for ${V}_{-4}$ with ${V}_{-6}$ on Λ; (

**d**) ${log}_{10}$ of the ${L}^{1}$ error for ${V}_{-5}$ with ${V}_{-6}$ on Λ; (

**e**) ${log}_{10}$ of the ${L}^{\infty}$ error for ${V}_{-4}$ with ${V}_{-6}$ on Λ; (

**f**) ${log}_{10}$ of the ${L}^{\infty}$ error for ${V}_{-5}$ with ${V}_{-6}$ on Λ.

Λ | “extra” | Terms | ${L}^{2}$ Error | ${L}^{1}$ Error | ${L}^{\infty}$ Error | |
---|---|---|---|---|---|---|

${V}_{-4}/{V}_{-6}$ | [3.75,6.25] | 3 | 326 | $4.66\times {10}^{-1}$ | $5.82\times {10}^{-1}$ | $6.61\times {10}^{-1}$ |

${V}_{-4}/{V}_{-6}$ | [3.50,6.50] | 4 | 360 | $2.97\times {10}^{-2}$ | $2.76\times {10}^{-2}$ | $4.42\times {10}^{-2}$ |

${V}_{-4}/{V}_{-6}$ | [3.25,7.00] | 5 | 410 | $6.95\times {10}^{-4}$ | $7.99\times {10}^{-4}$ | $7.45\times {10}^{-4}$ |

${V}_{-4}/{V}_{-6}$ | [3.00,7.00] | 6 | 428 | $3.09\times {10}^{-4}$ | $1.81\times {10}^{-4}$ | $5.98\times {10}^{-4}$ |

${V}_{-4}/{V}_{-6}$ | [2.50,7.50] | 7 | 494 | $3.65\times {10}^{-4}$ | $1.50\times {10}^{-4}$ | $5.98v{10}^{-4}$ |

${V}_{-5}/{V}_{-6}$ | [3.75,6.25] | 3 | 486 | $4.05\times {10}^{-2}$ | $2.78\times {10}^{-2}$ | $7.19\times {10}^{-2}$ |

${V}_{-5}/{V}_{-6}$ | [3.50,6.50] | 4 | 520 | $4.68\times {10}^{-4}$ | $3.14\times {10}^{-4}$ | $9.46\times {10}^{-4}$ |

${V}_{-5}/{V}_{-6}$ | [3.25,7.00] | 5 | 570 | $5.64\times {10}^{-5}$ | $2.13\times {10}^{-5}$ | $1.65\times {10}^{-4}$ |

${V}_{-5}/{V}_{-6}$ | [3.00,7.00] | 6 | 588 | $5.62\times {10}^{-5}$ | $1.81\times {10}^{-5}$ | $1.65\times {10}^{-4}$ |

${V}_{-5}/{V}_{-6}$ | [2.50,7.50] | 7 | 654 | $5.62\times {10}^{-5}$ | $1.78\times {10}^{-5}$ | $1.65\times {10}^{-4}$ |

**Figure 18.**The error at $t=50$ of the function Z from the non-linear ($\u03f5=0.01$) problem. The vertical axis is the ${log}_{10}$ of the error, the difference between the ${V}_{-3}:{V}_{-4}$ multi-scale results and the ${V}_{-4}$ control results (see Figure 15). The horizontal axis is the total number of ${a}_{j,k}$ and ${b}_{j,k}$ coefficients used for the y directional decomposition. (

**a**) ${log}_{10}$ of the ${L}^{2}$ error; (

**b**) ${log}_{10}$ of the ${L}^{1}$ error; (

**c**) ${log}_{10}$ of the ${L}^{\infty}$ error.

**Table 3.**Error from non-linear multi-scale Rossby wave results. All but the control use ${V}_{-3}$ and ${V}_{-4}$.

Λ | “extra” | Terms | Time | ${L}^{2}$ Error | ${L}^{1}$ Error | ${L}^{\infty}$ Error |
---|---|---|---|---|---|---|

$[3.5,8.5]$ | 4 | 168 | 3.66 | $2.60\times {10}^{-1}$ | $5.46\times {10}^{-1}$ | $3.07\times {10}^{-1}$ |

$[3.5,9.0]$ | 4 | 176 | 4.10 | $6.20\times {10}^{-2}$ | $1.33\times {10}^{-1}$ | $7.24\times {10}^{-2}$ |

$[3.5,9.5]$ | 4 | 184 | 4.67 | $2.55\times {10}^{-2}$ | $5.29\times {10}^{-2}$ | $3.40\times {10}^{-2}$ |

$[3.0,9.5]$ | 4 | 192 | 5.27 | $8.46\times {10}^{-3}$ | $2.07\times {10}^{-2}$ | $1.13\times {10}^{-2}$ |

$[2.5,9.5]$ | 4 | 200 | 6.16 | $4.18\times {10}^{-3}$ | $1.01\times {10}^{-2}$ | $3.79\times {10}^{-3}$ |

${V}_{-4}$ Control | 9.25 |

## 11. Conclusions

- First is the fact that both the large and small-scale systems should have adaptive decompositions. Due to how the systems are divided, there is no particular reason why the small-scale system in Λ could not follow any given adaptive scheme (that is compatible with an implicit time-discretization). Most of the changes in an adaptive scheme would be towards the finer resolution terms, which interact very little with the large-scale system. The real question is how to determine if a particularly tight concentration of fine resolution terms merits creating further small-scale systems.
- Giving the large-scale system an adaptive decomposition could be more complicated. Increasing the resolution of the large-scale system within Λ would probably be a bad idea. That would create duplicate terms (those that are calculated twice, once in each) to no net benefit. However, a few, isolated, small-scale calculations outside of Λ would obviously give greater accuracy, at limited expense. Again, the real question is when to decide that a set of small-scale terms would be best given its own small-scale system, and be calculated separately from the large-scale system.
- Our two dimensional example involves a sub-domain Λ that covers a small subset of the domain in the y direction, but covers the entire interval in the x direction. Basically, the example has two spatial dimensions, but the multi-scale method is only used in one of those dimensions. It would be instructive to test a problem requiring localization in two or more spatial dimensions. A turbulence related problem would be appropriate for testing purposes, or any problem from fluid dynamics that results in small-scale vortices.
- An analysis of the effect of our method on the stability of the underlying time-discretization. Preliminary testing was done in [23], showing a minimal effect on stability, but more is needed.
- Using multiple small-scale systems, either nested or discrete, as shown in Figure 2. If a problem requires high resolution in two regions, call them ${\Lambda}_{1}$ and ${\Lambda}_{2}$, it may be possible to calculate them separately, after the calculation of the large-scale system. The large-scale system could have, for example, 800 elements, with 400 small-scale elements and 200 large-scale in each of ${\Lambda}_{1}$ and ${\Lambda}_{2}$. Solving all of these together would involve 1200 elements. Solving them broken up into three systems would involve an 800 element system and two of 400. This should involve substantial savings.
- Using a different time step size for the small-scale system. Instead of calculating ${\mathbf{b}}^{n+1}$ after each time step of the large-scale system, we could calculate ${\mathbf{b}}^{n+\frac{1}{2}}$ then ${\mathbf{b}}^{n+1}$ (or ${\mathbf{b}}^{n+\frac{1}{4}}$, ${\mathbf{b}}^{n+\frac{1}{2}}$, ${\mathbf{b}}^{n+\frac{3}{4}}$, then ${\mathbf{b}}^{n+1}$). Doing so requires the intermediate large-scale vectors $\mathbf{a}$, which are relatively easy to calculate via interpolation. However, further experimentation is necessary to find how well this works, and what modifications may be necessary.
- Making the size of Λ, the resolutions used, and the number of “extra” terms adaptive. This requires analyzing the sources of error stemming from the boundaries of Λ and the lack of “extra” terms.

## Acknowledgements

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**MDPI and ACS Style**

McLaren, D.A.; Campbell, L.J.; Vaillancourt, R.
A Sequential, Implicit, Wavelet-Based Solver for Multi-Scale Time-Dependent Partial Differential Equations. *Axioms* **2013**, *2*, 142-181.
https://doi.org/10.3390/axioms2020142

**AMA Style**

McLaren DA, Campbell LJ, Vaillancourt R.
A Sequential, Implicit, Wavelet-Based Solver for Multi-Scale Time-Dependent Partial Differential Equations. *Axioms*. 2013; 2(2):142-181.
https://doi.org/10.3390/axioms2020142

**Chicago/Turabian Style**

McLaren, Donald A., Lucy J. Campbell, and Rémi Vaillancourt.
2013. "A Sequential, Implicit, Wavelet-Based Solver for Multi-Scale Time-Dependent Partial Differential Equations" *Axioms* 2, no. 2: 142-181.
https://doi.org/10.3390/axioms2020142