Comparison of Numerical Solvers for Anisotropic Diffusion Equations Arising in Plasma Physics

Abstract

This work is devoted to the comparison of numerical schemes to approximate anisotropic diffusion problems arising in tokamak plasma physics. We focus on the spatial approximation by using finite volume method and on the time discretization. This latter point is delicate since the use of explicit integrators leads to a severe restriction on the time step. Then, implicit and semi-implicit schemes are coupled to finite volumes space discretization and are compared for some classical problems relevant for magnetically confined plasmas. It appears that the semi-implicit approaches (using ARK methods or directional splitting) turn out to be the most efficient on the numerical results, especially when nonlinear problems are studied on refined meshes, using high order methods in space.

Appendix

1.1 Relations for Spatial Discretization

In this part, we recall some relations useful for the derivation of the numerical scheme. We consider one dimensional functions $f:=f(x)$ and $h:=h(x)$ and two-dimensional function $g:=g(x,y)$. A uniform one dimensional generic mesh is used: $z_{k+1/2}=z_{k-1/2}+\varDelta z$, for $k\in {{\mathbb {Z}}}$. We denote by $f_k$ the cell averaged quantity

$$\begin{aligned} f_k = \frac{1}{\varDelta z}\int _{z_{k-1/2}}^{z_{k+1/2}} f(z) dz. \end{aligned}$$

We recall the notations introduced above $C_i^x=[x_{i-1/2}, x_{i+1/2}]$ and $C_j^y=[y_{j-1/2}, y_{j+1/2}]$, with $\varDelta x$ and $\varDelta y$ the mesh size in the directions x and y so that the cell averaged writes

$$\begin{aligned} g_{i,j} = \frac{1}{\varDelta x\varDelta y}\int _{C_i^x}\int _{C_j^y} g(x, y) dxdy. \end{aligned}$$

The first one enables to express the product of the integral of two functions as a product of the integral of the function.

Proposition 7.1

Let us consider $I=\int _{z_{k-1/2}}^{z_{k+1/2}} f(z) h(z) dz$. A second order approximation of $I$ gives

$$\begin{aligned} \frac{1}{\varDelta z}\int _{z_{k-1/2}}^{z_{k+1/2}} f(z) h(z) dz = \frac{1}{\varDelta z}\int _{z_{k-1/2}}^{z_{k+1/2}} f(z) dz \frac{1}{\varDelta z}\int _{z_{k-1/2}}^{z_{k+1/2}} h(z) dz+ \mathcal{O}(\varDelta z^2) \end{aligned}$$

whereas a fourth order approximation of $I$ gives

$$\begin{aligned} \frac{1}{\varDelta z}\int _{z_{k-1/2}}^{z_{k+1/2}} f(z) h(z) dz&= \frac{1}{\varDelta z}\int _{z_{k-1/2}}^{z_{k+1/2}} f(z) dz \frac{1}{\varDelta z}\int _{z_{k-1/2}}^{z_{k+1/2}} h(z) dz\\&\quad +\, \frac{1}{48\varDelta z^2 }\left( \int _{z_{k+1/2}}^{z_{k+3/2}} f(z) dz \right. \\&\left. \quad -\, \int _{z_{k-3/2}}^{z_{k-1/2}} f(z) dz \right) \left( \int _{z_{k+1/2}}^{z_{k+3/2}} h(z) dz - \int _{z_{k-3/2}}^{z_{k-1/2}} h(z) dz \right) \\&\quad +\, \mathcal{O}(\varDelta z^4). \end{aligned}$$

The following proposition enables to express the face averaged values from the cell averaged values, up to order 4.

Proposition 7.2

A second order approximation of the face average of $f$ from cell averaged $f_k$ is

$$\begin{aligned} f(z_{k+1/2}) = \frac{1}{2} \left[ f_{k+1} + f_{k}\right] +\mathcal{O}(\varDelta z^2). \end{aligned}$$

whereas a fourth order approximation gives

$$\begin{aligned} f(z_{k+1/2}) = \frac{7}{12} \left[ f_{k+1} + f_{k}\right] - \frac{1}{12} \left[ f_{k+2} + f_{k-1}\right] +\mathcal{O}(\varDelta z^4). \end{aligned}$$

Then, we deduce, for a function $g=g(x,y)$,

$$\begin{aligned} \int _{C^y_j} g(x_{i+1/2},y) dy&= \frac{1}{2}(g_{i+1, j} + g_{i,j}) +\mathcal{O}(\varDelta x^2), \;\; \text{ for } \text{ the } \text{ second } \text{ order }\\&= \frac{7}{12} \left( g_{i+1,j} + g_{i,j} \right) - \frac{1}{12} \left( g_{i+2,j} + g_{i-1,j} \right) +\mathcal{O}(\varDelta x^4), \\&\text{ for } \text{ the } \text{ fourth } \text{ order } \end{aligned}$$

and

$$\begin{aligned} \int _{C^x_i} g(x,y_{j+1/2}) dx&= \frac{1}{2}(g_{i, j+1} + g_{i,j})+\mathcal{O}(\varDelta y^2), \;\; \text{ for } \text{ the } \text{ second } \text{ order }\\&= \frac{7}{12} \left( g_{i,j+1} + g_{i,j} \right) - \frac{1}{12} \left( g_{i,j+2} + g_{i,j-1} \right) +\mathcal{O}(\varDelta y^4), \\&\text{ for } \text{ the } \text{ fourth } \text{ order }. \end{aligned}$$

The following proposition enables to express the face averaged derivative from cell averaged values, up to order 4.

Proposition 7.3

For any function $f$, we have the following second order approximation

$$\begin{aligned} f'(z_{k+1/2}) = \frac{1}{\varDelta z }(f_{k+1}-f_k) + \mathcal{O}(\varDelta z^2). \end{aligned}$$

and the fourth order approximation

$$\begin{aligned} f'(z_{k+1/2}) = \frac{5}{4\varDelta z }\left[ f_{k+1} - f_{k} \right] -\frac{1}{12\varDelta z}\left[ f_{k+2} -f_{k-1}\right] +\mathcal{O}(\varDelta z ^4). \end{aligned}$$

Then, we deduce, for a function $g=g(x,y)$,

$$\begin{aligned} \int _{C^y_j} \partial _x g(x_{i+1/2},y) dy&= \frac{1}{\varDelta x}(g_{i+1,j}-g_{i,j}) + \mathcal{O}(\varDelta x^2), \;\; \text{ for } \text{ the } \text{ second } \text{ order }\\&= \frac{5}{4\varDelta x} \left( g_{i+1,j} - g_{i,j} \right) - \frac{1}{12\varDelta x} \left( g_{i+2,j} - g_{i-1,j} \right) + \mathcal{O}(\varDelta x^4), \\&\text{ for } \text{ the } \text{ fourth } \text{ order } \end{aligned}$$

and

$$\begin{aligned} \int _{C^x_i} \partial _y g(x,y_{j+1/2}) dx&= \frac{1}{\varDelta y}(g_{i,j+1}-g_{i,j})+ \mathcal{O}(\varDelta y^2), \;\; \text{ for } \text{ the } \text{ second } \text{ order }\\&= \frac{5}{4\varDelta y} \left( g_{i,j+1} -g_{i,j} \right) - \frac{1}{12\varDelta y} \left( g_{i,j+2} - g_{i,j-1} \right) + \mathcal{O}(\varDelta y^4), \\&\text{ for } \text{ the } \text{ fourth } \text{ order }. \end{aligned}$$

Finally, we present some relations useful for the numerical approximations of the cross derivatives.

Proposition 7.4

For a function $g=g(x,y)$, we get

$$\begin{aligned} \int _{C^x_i} \partial _x g(x,y_{j+1/2}) dx&= g(x_{i+1/2},y_{j+1/2})- g(x_{i-1/2},y_{j+1/2}) = (D_y D_x g)_{i,j}, \end{aligned}$$

and

$$\begin{aligned} \int _{C_j^y} \partial _y g(x_{i+1/2},y) dy&= g(x_{i+1/2},y_{j+1/2})- g(x_{i+1/2},y_{j-1/2}) = (D_x D_y g)_{i,j}. \end{aligned}$$

We used the notations

$$\begin{aligned} (D_x g)_{i,j}&= \frac{1}{\varDelta x}(g_{i+1,j}-g_{i,j})+\mathcal{O}(\varDelta x^2), \;\; \text{ for } \text{ the } \text{ second } \text{ order }\\&= \frac{7}{12\varDelta x}(g_{i+1,j}+g_{i,j}) -\frac{1}{12\varDelta x}(g_{i+2,j}+g_{i-1,j})+\mathcal{O}(\varDelta x^4), \\&\text{ for } \text{ the } \text{ fourth } \text{ order }. \end{aligned}$$

and

$$\begin{aligned} (D_y g)_{i,j}&= \frac{1}{\varDelta y}(g_{i,j+1}-g_{i,j})+\mathcal{O}(\varDelta y^2), \;\; \text{ for } \text{ the } \text{ second } \text{ order }\\&= \frac{7}{12\varDelta y}(g_{i,j+1}+g_{i,j}) -\frac{1}{12\varDelta y}(g_{i,j+2}+g_{i,j-1})+\mathcal{O}(\varDelta y^4), \\&\text{ for } \text{ the } \text{ fourth } \text{ order }, \end{aligned}$$

so that

$$\begin{aligned} (D_x D_y g)_{i,j}&= \frac{1}{4\varDelta x\varDelta y}(g_{i+1,j+1}-g_{i+1,j}+g_{i,j+1}-g_{i,j})+\mathcal{O}(\varDelta x^2)+\mathcal{O}(\varDelta y^2), \nonumber \\&\text{ for } \text{ the } \text{ second } \text{ order } \end{aligned}$$

(7.1)

$$\begin{aligned} (D_x D_y g)_{i,j}&= \frac{7}{12\varDelta x\varDelta y} \left[ \frac{7}{12}(g_{i+1,j+1}+g_{i+1,j})-\frac{1}{12} (g_{i+1,j+2}+g_{i+1,j-1})\right] \nonumber \\&+ \frac{7}{12\varDelta x\varDelta y} \left[ \frac{7}{12}(g_{i,j+1}+g_{i,j})-\frac{1}{12}(g_{i,j+2}+g_{i,j-1})\right] \nonumber \\&- \frac{1}{12\varDelta x\varDelta y} \left[ \frac{7}{12}(g_{i+2,j+1}+g_{i+2,j})-\frac{1}{12}(g_{i+2,j+2} +g_{i+2,j-1})\right] \nonumber \\&- \frac{1}{12\varDelta x\varDelta y} \left[ \frac{7}{12}(g_{i-1,j+1}+g_{i-1,j})-\frac{1}{12} (g_{i-1,j+2}+g_{i-1,j-1})\right] \nonumber \\&+ \mathcal{O}(\varDelta x^4)+\mathcal{O}(\varDelta y^4), \;\; \text{ for } \text{ the } \text{ fourth } \text{ order }. \end{aligned}$$

(7.2)

1.2 Linear Stability Analysis: Second Order

In this Appendix, the computations related to the linear stability analysis for the second order in space are detailed, for the SH and ARK schemes.

1.2.1 Semi-Implicit Scheme (SH)

The first step of the scheme writes

$$\begin{aligned} T^\star _{i,j}&= T^n_{i,j} + \frac{\varDelta t}{\varDelta x^2} b_{xx} \left[ T^\star _{i+1,j}-2T^\star _{i,j} + T^\star _{i-1,j}\right] \\&+\, \frac{\varDelta t}{\varDelta x\varDelta y} b_{xy} \left[ T^n_{i+1, j+1} -T^n_{i+1, j-1} -T^n_{i-1, j+1} +T^n_{i-1, j-1} \right] , \end{aligned}$$

whereas the second step is

$$\begin{aligned} T^{n+1}_{i,j}&= T^\star _{i,j} + \frac{\varDelta t}{\varDelta y^2} b_{yy} \left[ T^{n+1}_{i, j+1}-2T^{n+1}_{i,j} + T^{n+1}_{i, j-1}\right] \\&+ \,\frac{\varDelta t}{\varDelta x\varDelta y} b_{xy} \left[ T^\star _{i+1, j+1} -T^\star _{i+1, j-1} -T^\star _{i-1, j+1} +T^\star _{i-1, j-1} \right] . \end{aligned}$$

The Von Neuman analysis leads to the amplification factor for the first step

$$\begin{aligned} r_1 = \frac{1- \frac{\varDelta t}{\varDelta x^2} b_{xy} \sin (k_y\varDelta y) \sin (k_x\varDelta x)}{1+ \frac{4\varDelta t}{\varDelta x^2} b_{xx} \sin ^2 (k_x\varDelta x/2)}. \end{aligned}$$

In the same way, the amplification factor $r_2$ for the second step writes

$$\begin{aligned} r_2 = \frac{1- \frac{\varDelta t}{\varDelta x^2} b_{xy} \sin (k_y\varDelta y) \sin (k_x\varDelta x)}{ 1+ \frac{4\varDelta t}{\varDelta y^2} b_{yy} \sin ^2 (k_y\varDelta y/2) }, \end{aligned}$$

and then the total amplification factor is $r=r_1r_2$.

$$\begin{aligned} r := r(\text{ ncfl }, k_x \varDelta x, k_y \varDelta y) = \frac{(1- (\text{ ncfl }/2) b_{xy} A_x A_y\cos (k_x\varDelta x/2)\cos (k_y\varDelta y/2))^2 }{( 1+ (\text{ ncfl }/2) b_{xx}A_x^2)( 1+ (\text{ ncfl }/2) b_{yy}A_y^2)}, \end{aligned}$$

with $\varDelta t = \text{ ncfl } \frac{\varDelta x^2 }{4} = \text{ ncfl } \frac{\varDelta y^2 }{4}$ and $A_x=\sin (k_x\varDelta x/2)$ and $A_y=\sin (k_y\varDelta y/2)$. One can prove that $|r| \le 1$

$$\begin{aligned} |r|&\le \frac{(1+ (\text{ ncfl }/2) | b_{xy}| |A_x| |A_y|)^2 }{( 1+ (\text{ ncfl }/2) b_{xx}A_x^2)( 1+ (\text{ ncfl }/2) b_{yy}A_y^2)}\\&\le \frac{(1+ (\text{ ncfl }/2) \sqrt{b_{xx}}\sqrt{b_{yy}} |A_x| |A_y|)^2 }{( 1+ (\text{ ncfl }/2) b_{xx}A_x^2)( 1+ (\text{ ncfl }/2) b_{yy}A_y^2)}\\&= \frac{(1+ (\text{ ncfl }^2/4) b_{xx}b_{yy} A_x^2 A_y^2 +\text{ ncfl } \sqrt{b_{xx}}\sqrt{b_{yy}} |A_x| |A_y| ) }{( 1+ (\text{ ncfl }^2/4) b_{xx}b_{yy}A_x^2A_y^2 + (\text{ ncfl }/2) ( b_{xx}A_x^2+b_{yy}A_y^2))}\\&\le \frac{(1+ (\text{ ncfl }^2/4) b_{xx}b_{yy} A_x^2 A_y^2 +(\text{ ncfl }/2) (b_{xx}A_x^2 +b_{yy}A_y^2) ) }{( 1+ (\text{ ncfl }^2/4) b_{xx}b_{yy}A_x^2A_y^2 + (\text{ ncfl }/2) ( b_{xx}A_x^2+b_{yy}A_y^2))} = 1,\\ \end{aligned}$$

which proves the unconditional stability of the scheme.

1.2.2 Additive Runge–Kutta

We consider the second order spatial discretization coupled with a first order time discretization. The numerical scheme then reads

$$\begin{aligned}&T^{n+1}_{i,j} - \frac{\varDelta t\lambda }{\varDelta x^2}[T^{n+1}_{i+1,j}+T^{n+1}_{i,j+1}-4T^{n+1}_{i,j} +T^{n+1}_{i-1,j}+T^{n+1}_{i,j-1} ]\\&\quad = T^n_{i,j} + \frac{\varDelta t(b_{xx} -\lambda )}{\varDelta x^2}[T^{n}_{i+1,j}-2T^{n}_{i,j} +T^{n}_{i-1,j}]\\&\quad +\, \frac{\varDelta t(b_{yy} -\lambda )}{\varDelta x^2}[T^{n}_{i,j+1}-2T^{n}_{i,j} +T^{n}_{i,j-1}]\\&\quad +\,2\frac{\varDelta t b_{xy}}{4\varDelta x^2}(T^{n}_{i+1,j+1}-T^{n}_{i+1,j-1}-T^{n}_{i-1,j+1}+T^{n}_{i-1,j-1}). \end{aligned}$$

The Von Neuman analysis leads to the following amplification factor $r=r_1/r_2$ with $(\hbox {denoting ncfl}=(4\varDelta t)/\varDelta x^2)$

$$\begin{aligned} r_1&= 1 - \frac{4\varDelta t(b_{xx} -\lambda )}{\varDelta x^2}\sin ^2(k_x\varDelta x/2)- \frac{4\varDelta t(b_{yy} -\lambda )}{\varDelta x^2}\sin ^2(k_y\varDelta x/2) \\&\quad -\, 2\frac{\varDelta t b_{xy}}{\varDelta x^2}\sin (k_x\varDelta x)\sin (k_y\varDelta x),\\ r_2&= 1+ \frac{4\varDelta t\lambda }{\varDelta x^2}(\sin ^2(k_x\varDelta x/2)+\sin ^2(k_y\varDelta x/2)). \end{aligned}$$

Introducing $A_x=\sin (k_x\varDelta x/2)$ and $A_y=\sin (k_y\varDelta y/2)$, we get

$$\begin{aligned} \left| \frac{r_1}{r_2} \right| = \left| 1- \frac{\text{ ncfl } (b_{xx}A_x^2+ b_{yy} A_y^2- 2 b_{xy} A_xA_y \cos (k_x\varDelta x/2)\cos (k_y\varDelta x/2))}{1+\text{ ncfl } \lambda (A_x^2+A_y^2)}\right| . \end{aligned}$$

We now need to check the inequalities $-1<r_1/r_2<1$. The inequality $r_1/r_2<1$ is always ensured since the numerator is non negative:

$$\begin{aligned}&b_{xx}A_x^2+ b_{yy} A_y^2- 2 b_{xy} A_xA_y \cos (k_x\varDelta x/2)\cos (k_y\varDelta x/2)\\&\quad \ge b_{xx}A_x^2+ b_{yy} A_y^2- 2 b_{xy} A_xA_y , \end{aligned}$$

and

$$\begin{aligned} b_{xx}A_x^2+ b_{yy} A_y^2- 2 b_{xy} A_xA_y&= \frac{1}{b_{yy}} \left( b_{yy}b_{xx}A_x^2+ b^2_{yy} A_y^2- 2 b_{yy} b_{xy} A_xA_y\right) \\&\ge \frac{1}{b_{yy}} \left( b_{xy} A_x - b_{yy} A_y\right) ^2.\\ \end{aligned}$$

The inequality $-1< r_1/r_2$ rewrites as

$$\begin{aligned} \frac{\text{ ncfl } (b_{xx}A_x^2+ b_{yy} A_y^2- 2 b_{xy} A_xA_y \cos (k_x\varDelta x/2)\cos (k_y\varDelta x/2))}{1+\text{ ncfl } \lambda (A_x^2+A_y^2)}\le 2. \end{aligned}$$

But, with the following estimate on the numerator

$$\begin{aligned}&\left| b_{xx}A_x^2+ b_{yy} A_y^2- 2 b_{xy} A_xA_y \cos (k_x\varDelta x/2)\cos (k_y\varDelta x/2)) \right| \\&\quad \le \left| b_{xx}A_x^2+ b_{yy} A_y^2 + 2 \sqrt{|b_{xx}|}\sqrt{|b_{yy}|} A_xA_y \right| \\&\quad \le \left| b_{xx}A_x^2+ b_{yy} A_y^2 +b_{xx} b_{yy} A_x^2A_y^2 \right| \\&\quad \le 2 \left| b_{xx}A_x^2+ b_{yy} A_y^2 \right| , \end{aligned}$$

we deduce

$$\begin{aligned} \frac{\text{ ncfl } (b_{xx}A_x^2+ b_{yy} A_y^2)}{1+\text{ ncfl } \lambda (A_x^2+A_y^2)}\le 1, \end{aligned}$$

which is always true since $\lambda = \max (|X_1|, |X_2|)\ge \max (b_{xx}, b_{yy})$.

1.3 Linear Stability Analysis: Fourth Order

We detail in this part the computations of the linear stability analysis of the two time integrators we focus on: SH and ARK schemes.

1.3.1 Semi-Implicit Scheme (SH)

We write the numerical scheme of order four in space and couple with the semi-implicit scheme. We then define the first step of the scheme as (using the notation above)

$$\begin{aligned} T^\star _{i,j} = T^n_{i,j} + \frac{\varDelta t}{12\varDelta x^2} b_{xx} (D_{xx} T^\star )_{i,j} + \varDelta t b_{xy}(D_x D_y T^n)_{i,j}, \end{aligned}$$

whereas the second step writes

$$\begin{aligned} T^{n+1}_{i,j} = T^\star _{i,j} + \frac{\varDelta t}{12\varDelta y^2} b_{yy} (D_{yy} T^{n+1})_{i,j} + \varDelta t b_{xy}(D_x D_y T^\star )_{i,j}. \end{aligned}$$

where

$$\begin{aligned} (D_{xx} T)_{i,j}&= \left[ -T_{i+2,j} + 16T_{i+1,j} -30T_{i,j} + 16T_{i-1,j} - T_{i-2,j} \right] , \end{aligned}$$

(7.3)

$$\begin{aligned} (D_{yy} T)_{i,j}&= \left[ -T_{i,j+2} + 16T_{i,j+1} -30T_{i,j} + 16T_{i,j-1} - T_{i,j-2} \right] , \end{aligned}$$

(7.4)

and $(D_X D_y T)_{i,j}$ given by (7.2).

Following the Von Neumann analysis, we inject a plane wave solution $T(t, x, y)=r(t)\exp ^{-\text{ i }(k_x x+k_y y) }$ to get the amplification factor $r_1$ for the first step (with ncfl$=4 \varDelta t/\varDelta x^2$)

$$\begin{aligned} r_1(t)&= \frac{1 - \frac{\text{ ncfl }}{144}b_{xy} [ \sin (2k_y\varDelta y) - 8 \sin (k_y\varDelta y)] [\sin (2k_x\varDelta x) - 8 \sin (k_x\varDelta x)]}{ 1 + \frac{\text{ ncfl }}{48}b_{xx} [ 2\cos (2k_x\varDelta x) -32 \cos (k_x\varDelta x) +30 ] }\\&= \frac{1 - \frac{\text{ ncfl }}{144}b_{xy} [ \sin (2k_y\varDelta y) - 8 \sin (k_y\varDelta y)] [\sin (2k_x\varDelta x) - 8 \sin (k_x\varDelta x)]}{ 1 + \frac{\text{ ncfl }}{48}b_{xx} [ 4(\cos (k_x\varDelta x) -7)( \cos (k_x\varDelta x)-1) ] }, \end{aligned}$$

whereas the amplification factor $r_2$ for the second step is

$$\begin{aligned} r_2(t) = \frac{1 - \frac{\text{ ncfl }}{144}b_{xy} [ \sin (2k_y\varDelta y) - 8 \sin (k_y\varDelta y)] [\sin (2k_x\varDelta x) - 8 \sin (k_x\varDelta x)]}{ 1 + \frac{\text{ ncfl }}{48} b_{yy} [ 2\cos (2k_y\varDelta y) -32 \cos (k_y\varDelta y) +30 ] }. \end{aligned}$$

The amplification factor $r=r_1 r_2$ is plotted in Fig. 3 for different values of ncfl.

1.3.2 Additive Runge–Kutta

We write the numerical scheme of order four in space and couple with the penalisation technique. We then define the numerical scheme as

$$\begin{aligned} T^{n+1}_{i,j}&= T^n_{i,j} + \frac{\varDelta t}{12\varDelta x^2} (b_{xx}-\lambda )(D_{xx}T^n)_{i,j} + 2\varDelta t b_{xy}(D_x D_y T^n)_{i,j}\\&\quad +\, \frac{\varDelta t}{12\varDelta y^2} (b_{yy}-\lambda )(D_{xx}T^n)_{i,j} +\frac{\varDelta t \lambda }{\varDelta x^2}(D_{xx}T^{n+1})_{i,j} +\frac{\varDelta t \lambda }{\varDelta y^2}(D_{yy}T^{n+1})_{i,j}, \end{aligned}$$

where $(D_xx T)_{i,j}, (D_xx T)_{i,j}$ are given by (7.3) and (7.4), whereas $(D_x D_y T^n)_{i,j}$ is given by(7.2).

Following the Von Neumann analysis, we inject a plane wave solution $T(t, x, y)=r(t)\exp ^{-\text{ i }(k_x x+k_y y) }$ to get the amplification factor $r=r_1/r_2\, (\hbox {with}\, \hbox {ncfl}=4 \varDelta t/\varDelta x^2)$ with

$$\begin{aligned} r_1&= 1+\frac{\text{ ncfl } \lambda }{2} (- [\cos (2k_x\varDelta x) + \cos (2k_y\varDelta y)] \\&\quad +\, 16[\cos (k_x\varDelta x) + \cos (k_y\varDelta y)]-30)\\&\quad -\,\frac{\text{ ncfl } }{2} ( b_{xx}[-\cos (2k_x\varDelta x) + 16\cos (k_x\varDelta x)-15]\\&\quad +\, b_{yy}[ \cos (2k_y\varDelta y) + 16\cos (k_y\varDelta y)-15])\\&\quad -\, \frac{2\text{ ncfl } b_{xy}}{9}(\sin (2k_y\varDelta y) - 8\sin (k_y\varDelta y)][\sin (2k_x\varDelta x)-8\sin (k_x\varDelta x)]) \end{aligned}$$

and

$$\begin{aligned} r_2=1+\frac{\text{ ncfl } \lambda }{2}\Big (-[\cos (2k_x\varDelta x)+\cos (2k_y\varDelta y)] + 16[\cos (k_x\varDelta x)+\cos (k_y\varDelta y)]-30 \Big ). \end{aligned}$$

The amplification factor $r=r_1/r_2$ is plotted in Fig. 5 for different values of ncfl.