High-Order Finite-Difference Schemes for (Hyper-) Viscous Filtering on Non-Uniform Meshes

Abstract

In this study, the viscous filtering technique is extended to one-sided and biased finite-difference schemes for non-uniform meshes. The most attractive feature of this technique lies in its numerical stability despite the use of a purely explicit time advancement. This feature is well recovered for non-uniform meshes, making the approach as a simple and efficient alternative to the implicit time integration of the viscous term in the context of direct and large-eddy simulation. The rationale to develop generalized filter schemes is presented. After a validation based on the Burgers solution while using a refined mesh in the shock region, it is shown that a high-order formulation can be used to ensure both molecular and artificial dissipation for performing implicit LES of transitional boundary layer while relaxing drastically the time step constraint.

Appendices

Appendix 1: Viscous Filtering Without Splitting of the Diffusive Term (No Artificial Viscosity)

The filter is written as in Eq. (31) with $d_p=d_m=e_p=e_m=0$. The coefficients are solution of the system of equations (34).

The coefficients $M^p_j$, $M^p_j$ and $N_j$ in equations (34) are given by:

$$\begin{aligned} M^p_0= & {} 1\\ M^p_1= & {} \frac{1}{6} F^3 A^{31}_{i+1} \Delta s^5+\frac{1}{2} F^2 A^{21}_{i+1} \Delta s^3+F A^{11}_{i+1} \Delta s+1\\ M^p_2= & {} \frac{1}{3} F^3 A^{31}_{i+1} \Delta s^5+\frac{1}{3}F^3 A^{32}_{i+1} \Delta s^4+F^2 A^{21}_{i+1} \Delta s^3 + F^2 A^{22}_{i+1} \Delta s^2+2 F A^{11}_{i+1} \Delta s\\{} & {} + 2 F A^{12}_{i+1}+1\\ M^p_3= & {} \frac{1}{2}F^3 A^{31}_{i+1} \Delta s^5 + F^3 A^{32}_{i+1} \Delta s^4 + F^3 A^{33}_{i+1} \Delta s^3 +\frac{3}{2}F^2 A^{21}_{i+1} \Delta s^3 + 3 F^2 A^{22}_{i+1} \Delta s^2 \\{} & {} + 3 F^2 A^{23}_{i+1} \Delta s + 3 F A^{11}_{i+1} \Delta s + 6 F A^{12}_{i+1}+1\\ M^p_4= & {} \frac{2}{3}F^3 A^{31}_{i+1} \Delta s^5 + 2 F^3 A^{32}_{i+1} \Delta s^4 + 4 F^3 A^{33}_{i+1} \Delta s^3 + 2 F^2 A^{21}_{i+1} \Delta s^3 + 4 F^3 A^{34}_{i+1} \Delta s^2\\{} & {} + 6 F^2 A^{22}_{i+1} \Delta s^2 + 12 F^2 A^{23}_{i+1} \Delta s+4 F A^{11}_{i+1} \Delta s+12 F^2 A^{24}_{i+1}+12 F A^{12}_{i+1}+1 \\ M^p_5= & {} \frac{5}{6}F^3 A^{31}_{i+1} \Delta s^5 + \frac{10}{3}F^3 A^{32}_{i+1} \Delta s^4 + 10 F^3 A^{33}_{i+1} \Delta s^3 + \frac{5}{2}F^2 A^{21}_{i+1} \Delta s^3 + 20 F^3 A^{34}_{i+1} \Delta s^2 \\{} & {} + 10 F^2 A^{22}_{i+1} \Delta s^2 + 20 F^3 A^{35}_{i+1} \Delta s+ 30 F^2 A^{23}_{i+1} \Delta s+5 F A^{11}_{i+1} \Delta s+60 F^2 A^{24}_{i+1} \\{} & {} +20 F A^{12}_{i+1}+1 \\ M^p_6= & {} F^3 A^{31}_{i+1} \Delta s^5 + 5 F^3 A^{32}_{i+1} \Delta s^4 + 20 F^3 A^{33}_{i+1} \Delta s^3 + 3 F^2 A^{21}_{i+1} \Delta s^3 + 60 F^3 A^{34}_{i+1} \Delta s^2 \\{} & {} + 15 F^2 A^{22}_{i+1} \Delta s^2 + 120 F^3 A^{35}_{i+1} \Delta s + 60 F^2 A^{23}_{i+1} \Delta s + 6 F A^{11}_{i+1} \Delta s + 120 F^3 A^{36}_{i+1}\\{} & {} + 180 F^2 A^{24}_{i+1} + 30 F A^{12}_{i+1} + 1 \\ M^m_0= & {} 1\\ M^m_1= & {} \frac{1}{6}F^3 A^{31}_{i-1} \Delta s^5 + \frac{1}{2}F^2 A^{21}_{i-1} \Delta s^3 + F A^{11}_{i-1} \Delta s-1\\ M^m_2= & {} -\frac{1}{3}F^3 A^{31}_{i-1} \Delta s^5 + \frac{1}{3}F^3 A^{32}_{i-1} \Delta s^4 - F^2 A^{21}_{i-1} \Delta s^3 + F^2 A^{22}_{i-1} \Delta s^2 - 2 F A^{11}_{i-1} \Delta s\\{} & {} +2 F A^{12}_{i-1} + 1\\ M^m_3= & {} \frac{1}{2}F^3 A^{31}_{i-1} \Delta s^5 - F^3 A^{32}_{i-1} \Delta s^4 + F^3 A^{33}_{i-1} \Delta s^3 + \frac{3}{2}F^2 A^{21}_{i-1} \Delta s^3 - 3 F^2 A^{22}_{i-1} \Delta s^2 \\{} & {} + 3 F^2 A^{23}_{i-1} \Delta s + 3 F A^{11}_{i-1} \Delta s - 6 F A^{12}_{i-1} - 1\\ M^m_4= & {} -\frac{2}{3}F^3 A^{31}_{i-1} \Delta s^5 + 2 F^3 A^{32}_{i-1} \Delta s^4 - 4 F^3 A^{33}_{i-1} \Delta s^3 - 2 F^2 A^{21}_{i-1} \Delta s^3 + 4 F^3 A^{34}_{i-1} \Delta s^2 \\{} & {} + 6 F^2 A^{22}_{i-1} \Delta s^2 - 12 F^2 A^{23}_{i-1} \Delta s - 4 F A^{11}_{i-1} \Delta s + 12 F^2 A^{24}_{i-1} + 12 F A^{12}_{i-1} + 1\\ M^m_5= & {} \frac{5}{6}F^3 A^{31}_{i-1} \Delta s^5 - \frac{10}{3}F^3 A^{32}_{i-1} \Delta s^4 + 10 F^3 A^{33}_{i-1} \Delta s^3 + \frac{5}{2}F^2 A^{21}_{i-1} \Delta s^3 - 20 F^3 A^{34}_{i-1} \Delta s^2 \\{} & {} - 10 F^2 A^{22}_{i-1} \Delta s^2 + 20 F^3 A^{35}_{i-1} \Delta s + 30 F^2 A^{23}_{i-1} \Delta s + 5 F A^{11}_{i-1} \Delta s - 60 F^2 A^{24}_{i-1}\\{} & {} - 20 F A^{12}_{i-1}-1\\ M^m_6= & {} -F^3 A^{31}_{i-1} \Delta s^5 + 5 F^3 A^{32}_{i-1} \Delta s^4 - 20 F^3 A^{33}_{i-1} \Delta s^3 - 3 F^2 A^{21}_{i-1} \Delta s^3 + 60 F^3 A^{34}_{i-1} \Delta s^2 \\{} & {} + 15 F^2 A^{22}_{i-1} \Delta s^2 - 120 F^3 A^{35}_{i-1} \Delta s - 60 F^2 A^{23}_{i-1} \Delta s - 6 F A^{11}_{i-1} \Delta s + 120 F^3 A^{36}_{i-1}\\{} & {} + 180 F^2 A^{24}_{i-1} + 30 F A^{12}_{i-1} + 1\\ N_0= & {} 1\\ N_1= & {} \frac{1}{6}F^3 A^{31}_i \Delta s^5+\frac{1}{2}F^2 A^{21}_i \Delta s^3+ F A^{1}_i \Delta s\\ N_2= & {} \frac{1}{3}F^3 A^{32}_i \Delta s^4+ F^2 A^{22}_i \Delta s^2+2 F A^{12}_i\\ N_3= & {} F^3 A^{33}_i \Delta s^3+3 F^2 A^{23}_i \Delta s\\ N_4= & {} 4 F^3 A^{34}_i \Delta s^2+12 F^2 A^{24}_i\\ N_5= & {} 20 F^3 A^{35}_i \Delta s\\ N_6= & {} 120 F^3 A^{36}_i\\ \end{aligned}$$

From Eq. (28), the successive derivatives can be calculated using the chain rule as

$$\begin{aligned} \frac{\partial {u}}{\partial {t}}= & {} \nu \left( g^2\frac{\partial ^2 {u}}{\partial {s}^2}+g g'\frac{\partial {u}}{\partial {s}}\right) \\ \frac{\partial ^2 {u}}{\partial {t}^2}= & {} \nu \left( g^2\frac{\partial ^2 {}}{\partial {s}^2}\left( \frac{\partial {u}}{\partial {t}}\right) +g g' \frac{\partial {}}{\partial {s}}\left( \frac{\partial {u}}{\partial {t}}\right) \right) \\= & {} \nu ^2\biggl ( \bigr . g^4 \frac{\partial ^{4} {u}}{\partial {s}^{4}} +6 g^3 g' \frac{\partial ^{3} {u}}{\partial {s}^{3}} +4 g^3 g^{(2)} \frac{\partial ^{2} {u}}{\partial {s}^{2}} +7 g^2 g'^2 \frac{\partial ^{2} {u}}{\partial {s}^{2}}\\{} & {} +g^3 g^{(3)} \frac{\partial {u}}{\partial {s}} +4 g^2 g' g^{(2)} \frac{\partial {u}}{\partial {s}} +g g'^3 \frac{\partial {u}}{\partial {s}} \biggl .\biggr ) \\ \frac{\partial ^{3} {u}}{\partial {t}^{3}}= & {} \nu \left( g^2\frac{\partial ^2 {}}{\partial {s}^2}\left( \frac{\partial ^2 {u}}{\partial {t}^2}\right) +g g'\frac{\partial {}}{\partial {s}}\left( \frac{\partial ^2 {u}}{\partial {t}^2}\right) \right) \\= & {} \nu ^3 \biggl ( \biggr . g^6 \frac{\partial ^{6} {u}}{\partial {s}^{6}} +15 g^5 g' \frac{\partial ^{5} {u}}{\partial {s}^{5}} +20 g^5 g^{(2)} \frac{\partial ^{4} {u}}{\partial {s}^{4}} +65 g^4 g'^2 \frac{\partial ^{4} {u}}{\partial {s}^{4}} + 15 g^5 g^{(3)} \frac{\partial ^{3} {u}}{\partial {s}^{3}}\\{} & {} + 120 g^4 g' g^{(2)} \frac{\partial ^{3} {u}}{\partial {s}^{3}}+90 g^3 g'^3 \frac{\partial ^{3} {u}}{\partial {s}^{3}}+6 g^5 g^{(4)} \frac{\partial ^{2} {u}}{\partial {s}^{2}}+57 g^4 g' g^{(3)} \frac{\partial ^{2} {u}}{\partial {s}^{2}}\\{} & {} + 34 g^4 {g^{(2)}}^2 \frac{\partial ^{2} {u}}{\partial {s}^{2}} +146 g^3 g'^2 g^{(2)} \frac{\partial ^{2} {u}}{\partial {s}^{2}} +31 g^2 g'^4 \frac{\partial ^{2} {u}}{\partial {s}^{2}} + g^5 g^{(5)} \frac{\partial {u}}{\partial {s}}\\{} & {} + 11 g^4 g' g^{(4)} \frac{\partial {u}}{\partial {s}} +15 g^4 g^{(2)} g^{(3)} \frac{\partial {u}}{\partial {s}} +32 g^3 g'^2 g^{(3)} \frac{\partial {u}}{\partial {s}}\\{} & {} + 34 g^3 g' {g^{(2)}}^2 \frac{\partial {u}}{\partial {s}} +26 g^2 g'^3 g^{(2)} \frac{\partial {u}}{\partial {s}} +g g'^5 \frac{\partial {u}}{\partial {s}} \biggl .\biggr ) \end{aligned}$$

which leads to the following expressions for the coefficients $A^{lm}$

$$\begin{aligned} A^{11}= & {} g g'\\ A^{12}= & {} {{g}^{2}}\\ A^{21}= & {} {{g}^{3}} g^{(3)} +4 {{g}^{2}} g' g^{(2)} +g {{g' }^{3}}\\ A^{22}= & {} 4 {{g}^{3}} g^{(2)} +7 {{g}^{2}} {{g' }^{2}}\\ A^{23}= & {} 6 {{g}^{3}} g' \\ A^{24}= & {} {{g}^{4}}\\ A^{31}= & {} {{g}^{5}} g^{(5)} +11 {{g}^{4}} g' g^{(4)} +\left( 15 {{g}^{4}} g^{(2)} +32 {{g}^{3}} {{g' }^{2}}\right) g^{(3)} +34 {{g}^{3}} g' {{g^{(2)} }^{2}}+26 {{g}^{2}} {{g' }^{3}} g^{(2)} +g {{g' }^{5}}\\ A^{32}= & {} 6 {{g}^{5}} g^{(4)} +57 {{g}^{4}} g' g^{(3)} +34 {{g}^{4}} {{g^{(2)} }^{2}}+146 {{g}^{3}} {{g' }^{2}} g^{(2)} +31 {{g}^{2}} {{g' }^{4}}\\ A^{33}= & {} 15 {{g}^{5}} g^{(3)} +120 {{g}^{4}} g' g^{(2)} +90 {{g}^{3}} {{g' }^{3}}\\ A^{34}= & {} 20 {{g}^{5}} g^{(2)} +65 {{g}^{4}} {{g' }^{2}}\\ A^{35}= & {} 15 {{g}^{5}} g'\\ A^{36}= & {} {{g}^{6}} \end{aligned}$$

Appendix 2: Viscous Filtering with Splitting of the Diffusive Term (No Artificial Viscosity)

The filter is written as in Eq. (31). The coefficients are solution of the system of equations (34) with $d_p=d_m=e_p=e_m=0$, and with:

$$\begin{aligned} M^p_0= & {} 1 \\ M^p_1= & {} 1 \\ M^p_2= & {} \frac{1}{3}F^3 A^{32}_{i+1} \Delta s^4 + F^2 A^{22}_{i+1} \Delta s^2 + 2 F A^{12}_{i+1} + 1 \\ M^p_3= & {} F^3 A^{32}_{i+1} \Delta s^4 + F^3 A^{33}_{i+1} \Delta s^3 + 3 F^2 A^{22}_{i+1} \Delta s^2 + 3 F^2 A^{23}_{i+1} \Delta s + 6 F A^{12}_{i+1} + 1\\ M^p_4= & {} 2 F^3 A^{32}_{i+1} \Delta s^4 + 4 F^3 A^{33}_{i+1} \Delta s^3 + 4 F^3 A^{34}_{i+1} \Delta s^2 + 6 F^2 A^{22}_{i+1} \Delta s^2 + 12 F^2 A^{23}_{i+1} \Delta s \\{} & {} + 12 F^2 A^{24}_{i+1} + 12 F A^{12}_{i+1} + 1 \\ M^p_5= & {} \frac{10}{3}F^3 A^{32}_{i+1} \Delta s^4 + 10 F^3 A^{33}_{i+1} \Delta s^3 + 20 F^3 A^{34}_{i+1} \Delta s^2 + 10 F^2 A^{22}_{i+1} \Delta s^2 + 20 F^3 A^{35}_{i+1} \Delta s \\{} & {} + 30 F^2 A^{23}_{i+1} \Delta s + 60 F^2 A^{24}_{i+1}+ 20 F A^{12}_{i+1} + 1 \\ M^p_6= & {} 5 F^3 A^{32}_{i+1} \Delta s^4 + 20 F^3 A^{33}_{i+1} \Delta s^3 + 60 F^3 A^{34}_{i+1} \Delta s^2 + 15 F^2 A^{22}_{i+1} \Delta s^2 + 120 F^3 A^{35}_{i+1} \Delta s \\{} & {} + 60 F^2 A^{23}_{i+1} \Delta s + 120 F^3 A^{36}_{i+1} + 180 F^2 A^{24}_{i+1} + 30 F A^{12}_{i+1} + 1 \\ M^m_0= & {} 1 \\ M^m_1= & {} -1 \\ M^m_2= & {} \frac{1}{3} F^3 A^{32}_{i-1} \Delta s^4 + F^2 A^{22}_{i-1} \Delta s^2 + 2 F A^{12}_{i-1} + 1 \\ M^m_3= & {} -F^3 A^{32}_{i-1} \Delta s^4 + F^3 A^{33}_{i-1} \Delta s^3 - 3 F^2 A^{22}_{i-1} \Delta s^2 + 3 F^2 A^{23}_{i-1} \Delta s - 6 F A^{12}_{i-1} - 1 \\ M^m_4= & {} 2 F^3 A^{32}_{i-1} \Delta s^4 - 4 F^3 A^{33}_{i-1} \Delta s^3 + 4 F^3 A^{34}_{i-1} \Delta s^2 + 6 F^2 A^{22}_{i-1} \Delta s^2 - 12 F^2 A^{23}_{i-1} \Delta s \\{} & {} + 12 F^2 A^{24}_{i-1} + 12 F A^{12}_{i-1} + 1 \\ M^m_5= & {} -\frac{10}{3} F^3 A^{32}_{i-1} \Delta s^4 + 10 F^3 A^{33}_{i-1} \Delta s^3 - 20 F^3 A^{34}_{i-1} \Delta s^2 - 10 F^2 A^{22}_{i-1} \Delta s^2 + 20 F^3 A^{35}_{i-1} \Delta s \\{} & {} + 30 F^2 A^{23}_{i-1} \Delta s - 60 F^2 A^{24}_{i-1} - 20 F A^{12}_{i-1} - 1 \\ M^m_6= & {} 5 F^3 A^{32}_{i-1} \Delta s^4 - 20 F^3 A^{33}_{i-1} \Delta s^3 + 60 F^3 A^{34}_{i-1} \Delta s^2 + 15 F^2 A^{22}_{i-1} \Delta s^2 - 120 F^3 A^{35}_{i-1} \Delta s \\{} & {} - 60 F^2 A^{23}_{i-1} \Delta s + 120 F^3 A^{36}_{i-1} + 180 F^2 A^{24}_{i-1} + 30 F A^{12}_{i-1} + 1 \\ N_0= & {} 1\\ N_1= & {} 0\\ N_2= & {} \frac{1}{3}F^3 A^{32}_i \Delta s^4 + F^2 A^{22}_i \Delta s^2 + 2 F A^{12}_i\\ N_3= & {} F^3 A^{33}_i \Delta s^3 + 3 F^2 A^{23}_i \Delta s\\ N_4= & {} 4 F^3 A^{34}_i \Delta s^2 + 12 F^2 A^{24}_i\\ N_5= & {} 20 F^3 A^{35}_i \Delta s\\ N_6= & {} 120 F^3 A^{36}_i\\ \end{aligned}$$

From Eq. (35), the successive derivatives can be calculated using the chain rule as

$$\begin{aligned} \frac{\partial {u}}{\partial {t}}= & {} \nu g^2\frac{\partial ^2 {u}}{\partial {s}^2}\\ \frac{\partial ^2 {u}}{\partial {t}^2}= & {} \nu g^2\frac{\partial ^2 {}}{\partial {s}^2}\left( \frac{\partial {u}}{\partial {t}}\right) \\= & {} \nu ^2\left( g^4 \frac{\partial ^{4} {u}}{\partial {s}^{4}} +4 g^3 g' \frac{\partial ^{3} {u}}{\partial {s}^{3}} +\left( 2 g^3 g^{(2)} +2 g^2 {{g' }^{2}}\right) \frac{\partial ^{2} {u}}{\partial {s}^{2}} \right) \\ \frac{\partial ^{3} {u}}{\partial {t}^{3}}= & {} \nu g^2\frac{\partial ^2 {}}{\partial {s}^2}\left( \frac{\partial ^2 {u}}{\partial {t}^2}\right) \\= & {} \nu ^3\biggl (\biggr . g^6 \frac{\partial ^{6} {u}}{\partial {s}^{6}} +12 g^5 g' \frac{\partial ^{5} {u}}{\partial {s}^{5}} +\left( 14 g^5 g^{(2)} +38 g^4 {{g' }^{2}}\right) \frac{\partial ^{4} {u}}{\partial {s}^{4}}\\{} & {} +\left( 8 g^5 g^{(3)} +56 g^4 g' g^{(2)} +32 g^3 {{g' }^{3}}\right) \frac{\partial ^{3} {u}}{\partial {s}^{3}}\\{} & {} +\left( 2 g^5 g^{(4)} +16 g^4 g' g^{(3)} +10 g^4 {{g^{(2)} }^{2}} +32 g^3 {{g' }^{2}} g^{(2)} +4 g^2 {{g' }^{4}}\right) \frac{\partial ^{2} {u}}{\partial {s}^{2}} \biggl .\biggr )\\ \end{aligned}$$

which leads to the following expressions for the coefficients $A^{lm}$

$$\begin{aligned} A^{12}= & {} g^2 \\ A^{22}= & {} 2 g^3 g^{(2)} +2 g^2 {{g' }^{2}} \\ A^{23}= & {} 4 g^3 g' \\ A^{24}= & {} g^4 \\ A^{32}= & {} 2 g^5 g^{(4)} +16 g^4 g' g^{(3)} +10 g^4 {{g^{(2)} }^{2}}+32 g^3 {{g' }^{2}} g^{(2)} +4 g^2 {{g' }^{4}} \\ A^{33}= & {} 8 g^5 g^{(3)} +56 g^4 g' g^{(2)} +32 g^3 {{g' }^{3}} \\ A^{34}= & {} 14 g^5 g^{(2)} +38 g^4 {{g' }^{2}} \\ A^{35}= & {} 12 g^5 g' \\ A^{36}= & {} g^6\\ \end{aligned}$$

Appendix 3: Additional Order Equation for the Filters with Artificial Viscosity

The coefficients $M^p_{7}$, $M^m_{7}$ and $N_7$ in Eq. (45) are given by:

$$\begin{aligned} M^p_7= & {} 7 F^3 A^{32}_{i+1} \Delta s^4 + 35 F^3 A^{33}_{i+1} \Delta s^3 + 140 F^3 A^{34}_{i+1} \Delta s^2 + 21 F^2 A^{22}_{i+1} \Delta s^2 + 420 F^3 A^{35}_{i+1} \Delta s \\{} & {} + 105 F^2 A^{23}_{i+1} \Delta s + 840 F^3 A^{36}_{i+1} + 420 F^2 A^{24}_{i+1} + 42 F A^{12}_{i+1} + 1\\ M^m_7= & {} -7 F^3 A^{32}_{i-1} \Delta s^4 + 35 F^3 A^{33}_{i-1} \Delta s^3 - 140 F^3 A^{34}_{i-1} \Delta s^2 - 21 F^2 A^{22}_{i-1} \Delta s^2 + 420 F^3 A^{35}_{i-1} \Delta s \\{} & {} + 105 F^2 A^{23}_{i-1} \Delta s - 840 F^3 A^{36}_{i-1} - 420 F^2 A^{24}_{i-1} - 42 F A^{12}_{i-1} - 1\\ N_7= & {} 0 \end{aligned}$$

Appendix 4: One-Sided Viscous Filtering (with Splitting of the Diffusive Term)

The filter is written as in Eq. (27). The coefficients are solution of the system (46) with:

$$\begin{aligned} M^p_0= & {} 1 \\ M^p_1= & {} 1 \\ M^p_2= & {} F^2 A^{22}_2 \Delta s^2+2 F A^{12}_2 + 1\\ M^p_3= & {} 3F^2A^{22}_2 \Delta s^2+3 F^2 A^{23}_2 \Delta s+6 F A^{12}_2 + 1 \\ M^p_4= & {} 6 F^2 A^{22}_2 \Delta s^2+12 F^2 A^{23}_2 \Delta s+12 F^2 A^{24}_2+12 F A^{12}_2 + 1 \\ N_0= & {} 1 \\ N_1= & {} 0 \\ N_2= & {} F^2 A^{22}_1 \Delta s^2+2 F A^{12}_1\\ N_3= & {} 3 F^2 A^{23}_1 \Delta s\\ N_4= & {} 12 F^2 A^{24}_1\\ \end{aligned}$$

with $A^{lm}_i$ at points $i=1,2$ defined as in Appendix 2.

Appendix 5: Numerical Stability Analysis

This appendix briefly summarises the stability condition for each scheme developed in the paper. For uniform meshes, the stability is simply analysed in the sense of von Neumann, using the condition $|T|\leqslant 1$, where T is the transfer function of the filter (6), given by

$$T(\kappa \Delta x)=\frac{a+b\cos (\kappa \Delta x)+c\cos (2\kappa \Delta x)+d\cos (3\kappa \Delta x)+e\cos (4\kappa \Delta x)}{1+\alpha \cos (\kappa \Delta x)}$$

For stretched meshes, the stability is analysed by calculating the eigenvalues of the system corresponding to the application of the filter $A\tilde{U}=BU$, where the vectors $\tilde{U}$ and U contain the values $\tilde{u}_i$ and $u_i$ at each grid point, and the matrices A and B contain the filter coefficients at each grid point (A is tridiagonal with coefficients $\alpha _m$, 1, $\alpha _p$ and B is a 9-diagonal matrix containing the coefficients $a, b_{p/m}, c_{p/m}, d_{p/m}, e_{p/m}$). The generalised eigenvalue problem can simply be written as

$$\begin{aligned} B\Phi =\lambda A\Phi \end{aligned}$$

and the scheme is stable if $\max |\lambda |\leqslant 1$.

For uniform meshes, the stability conditions for schemes $({\text{o6}})$ and $(T_c,{\text{o6}})$ have been provided by Lamballais et al. (2021), with $F\lesssim 0.15$ and $F\lesssim 4.17$, respectively. It has been shown in particular that the restriction on F for the scheme $({\text{o6}})$ is associated with a singular behaviour at $F\approx 0.158$. While the restriction for the scheme $(T_c,{\text{o6}})$ is found independent of $\nu _0/\nu$, it is found that the stability condition for the scheme $(T_m,T_c,{\text{o6}})$ is dependent on $\nu _0/\nu$ and $c_1$, as shown by Fig. 7-left, in which examples of iso-contour $|T|=1$ are plotted for different values of $\nu _0/\nu$ in the plane $(\kappa \Delta x - F)$, and where an unstable range is observed at intermediate Fourier numbers (for instance, the unstable range is $0.02\lesssim F\lesssim 0.6$ when $\nu _0/\nu =25$, as shown in the figure). Although a weak dependence is also observed with the scheme $(T_m,T_c,\alpha _0,{\text{o6}})$, the stability condition remains approximately $F\lesssim 1$. By showing the minimal $F=F_{stab}$ at which $|T|>1$, Fig. 7-right summarises the stability limits $F\leqslant F_{stab}$ for each scheme, as a function of $\nu _0/\nu$ and compares these restrictions with the stability condition of the conventional explicit scheme AB2, given by Eq. (22).

Fig. 7

Stability conditions for the schemes on uniform mesh. Left: iso-contour $|T|=1$ of the scheme $(T_m,T_c,o6)$ for different values of $\nu _0/\nu$, and for $c_1=c_{SVV}/2\approx 0.22$, in the plane $(\kappa \Delta x - F)$; Right: stability condition for each scheme as a function of $\nu _0/\nu$. For schemes $(T_m,T_c,o6)$ and $(T_m,T_c,\alpha _0,o6)$, continuous lines correspond to $c_1=c_{SVV}\approx 0.44$, dash lines to $c_1=c_{SVV}/2$, dotted lines to $c_1=c_{SVV}/8$ and dash-dotted lines to $c_1=c_{SVV}/64$

For stretched meshes, Fig. 8 shows the regions $\max |\lambda |>1$ in the plane $(F_{max}-\beta )$ where $F_{max}=\frac{\nu \Delta t}{\Delta x_{min}}$ and $\beta$ is the stretching parameter defining the mesh, together with the stability condition of each corresponding scheme on a uniform mesh, which is indicated by the dash red line. In this figure, the mesh is chosen refined at the centre, as for the Burgers equation in Sect. 4, with a number of points $N=512$, and the parameters $\nu _0/\nu =25$ and $c_1=c_{SVV}/2$ are chosen for the schemes including artificial viscosity. It can first be observed that the behaviour for uniform meshes is recovered for every scheme when $\beta$ is large. In particular, singular behaviours are observed close to $F\approx 0.15$ for the scheme $({\text{o6}})$ and in the range $0.02\lesssim F\lesssim 0.6$ for the scheme $(T_m,T_c,{\text{o6}})$. At lower values of $\beta$ (higher stretching), these singular behaviours are observed in an interval of $F_{max}$ which becomes wider when $\beta$ decreases. Considering that $F_x$ is the quantity on which the stability depends, this interval of unstable $F_{max}$ can be explained by the fact that a given mesh ($\beta$ fixed) involves an interval of Fourier numbers $F_{min}\le F_x \le F_{max}$. Assuming that the stability can be deduced locally from the stability of the schemes on uniform mesh, the schemes are expected to exhibit a singular behaviour when the Fourier numbers at which the schemes on uniform mesh are singular are in this interval. The iso-line $F_{min}=0.15$ is superimposed on Fig. 8 for the scheme $({\text{o6}})$ (green dash line) and it can be seen that the region where singular behaviours are observed is well enclosed by the lines $F_{max}=0.15$ and $F_{min}=0.15$, which tends to confirm this explanation, in particular the local character of the singular behaviours. The same observations can be made for the scheme $(T_m,T_c,{\text{o6}})$, where the green dash line is the iso-line $F_{min}=0.6$, and corresponds to the unstable range on a uniform mesh $0.02\lesssim F\lesssim 0.6$ for this scheme. Although not shown here, these ranges of $F_{max}$ leading to singular behaviours have been observed independently of the number of points N considered in the mesh. Considering the lowest $F_{max}$ at which the scheme becomes unstable, this analysis confirms that the stability condition is set by $F_{max}$ for these two schemes, as explained in Sect. 3.3.

Fig. 8

$\max |\lambda |$ on periodic stretched mesh, refined at the centre ($N=512$). Regions where $\max |\lambda |>1$ are coloured in grey and iso-lines $\max |\lambda |=1+\epsilon$ are plotted in black with $\epsilon =10^{-5}$. Red dash lines indicates the stability condition of each corresponding scheme on uniform mesh

Concerning the schemes $(T_c,{\text{o6}})$ and $(T_m,T_c,\alpha _0,{\text{o6}})$, it can be seen that the minimum $F_{max}$ at which the schemes become unstable is also almost independent of $\beta$. Although not shown here for the sake of conciseness, we have observed a small dependency of these results to the number of points for the scheme $(T_c,{\text{o6}})$, the stability condition being somewhat more restrictive for low values of $\beta$, when N is decreased (the minimum unstable $F_{max}$ is reduced to $\sim$ 2 at low values of $\beta$ when the number of points is reduced to $N=64$, which remains well above the restrictions typically encountered in DNS/LES due to the CFL condition). This dependency disappears when N is sufficiently large (when N is increased, the minimum unstable $F_{max}\sim 4$ remains independent of $\beta$).

Except for this dependency which appears to be significant only for the scheme $(T_c,{\text{o6}})$ with a small number of points, this analysis therefore confirms the general rule of thumb given in Sect. 3.3, namely that, for a sufficiently large number of points, the stable condition is given by the maximum Fourier number $F_{\max}=\nu \Delta t/\Delta x_{\min}^2$ while referring to the critical value for the counterpart scheme on a uniform mesh, as indicated by the red dash lines in Fig. 8.