An Explicit Exponential Integrator Based on Faber Polynomials and its Application to Seismic Wave Modeling

Abstract

Exponential integrators have been applied successfully in several physics-related differential equations. However, their application in hyperbolic systems with absorbing boundaries, like the ones arising in seismic imaging, still lacks theoretical and experimental investigations. The present work conducts an in-depth study of exponential integration using Faber polynomials, consisting of a generalization of a well-known exponential method that uses Chebyshev polynomials. This allows solving non-symmetric operators that emerge from classic seismic wave propagation problems with absorbing boundaries. Theoretical as well as numerical results are presented for Faber approximations. One of the theoretical contributions is the proposal of a sharp bound for the approximation error of the exponential of a normal matrix. We also show the practical importance of determining an optimal ellipse encompassing the full spectrum of the discrete operator to ensure and enhance the convergence of the Faber exponential series. Furthermore, based on estimates of the spectrum of the discrete operator of the wave equations with a widely used absorbing boundary method, we numerically investigate the stability, dispersion, convergence, and computational efficiency of the Faber exponential scheme. Overall, we conclude that the method is suitable for seismic wave problems and can provide accurate results with large time step sizes, with computational efficiency increasing with the increase of the approximation degree.

A Appendix

1.1 Appendix A.1: Continuous Framework

We formally write the equations formulations, with PML conditions, used throughout the work. For 1D the domain $\varOmega =[a_1,a_2]$ is an interval and for 2D $\varOmega =[0,a_2]\times [0,b_2]$ is considered a square. The particular values of $a_1$ and $a_2$ are fixed in the numerical tests.

One dimensional acoustic waves with PML:

• 1. Using second order spatial derivatives (2SD)

  $$\begin{aligned} \frac{\partial u}{\partial t}(x,t)&=v(x,t), \end{aligned}$$

  (36)
  $$\begin{aligned} \frac{\partial v}{\partial t}(x,t)&=-\beta _x(x)v(x,t)+c^2(x)\left( \frac{\partial ^2 u}{\partial x^2}(x,t)+\frac{\partial w}{\partial x}(x,t)\right) +f(x,t), \end{aligned}$$

  (37)
  $$\begin{aligned} \frac{\partial w}{\partial t}(x,t)&=-\beta _x(x)\left( w(x,t)+\frac{\partial u}{\partial x}(x,t)\right) . \end{aligned}$$

  (38)

• 2. Using only first order spatial derivatives [13] (1SD)

  $$\begin{aligned} \frac{\partial u}{\partial t}(x,t)&=c^2(x)\left( \frac{\partial v}{\partial x}(x,t) - w(x,t)\right) +\int \limits _{t_0}^t f(x,s)ds, \end{aligned}$$

  (39)
  $$\begin{aligned} \frac{\partial v}{\partial t}(x,t)&=-\beta _x(x)v(x,t)+\frac{\partial u}{\partial x}(x,t), \end{aligned}$$

  (40)
  $$\begin{aligned} \frac{\partial w}{\partial t}(x,t)&=\beta _x(x)\left( -w(x,t)+\frac{\partial v}{\partial x}(x,t)\right) . \end{aligned}$$

  (41)

Two dimension acoustic waves with PML:

• 1. Using second order spatial derivatives (2SD)

  $$\begin{aligned} \frac{\partial u}{\partial t}(x,y,t)&=v(x,y,t),\end{aligned}$$

  (42)
  $$\begin{aligned} \frac{\partial v}{\partial t}(x,y,t)&=-\big (\beta _x(x)+\beta _y(y)\big )v(x,y,t)-\beta _x(x)\beta _y(y)u(x,y,t)\nonumber \\&\quad +c^2(x,y)\left( \frac{\partial ^2 u}{\partial x^2}(x,y,t)+\frac{\partial ^2 u}{\partial y^2}(x,y,t)+\frac{\partial w_x}{\partial x}(x,y,t)\right. \nonumber \\&\quad \left. +\frac{\partial w_y}{\partial y}(x,y,t)\right) +f(x,y,t), \end{aligned}$$

  (43)
  $$\begin{aligned} \frac{\partial w_x}{\partial t}(x,y,t)&=-\beta _x(x)w_x(x,y,t)+(\beta _y(y)-\beta _x(x))\frac{\partial u}{\partial x}(x,y,t), \end{aligned}$$

  (44)
  $$\begin{aligned} \frac{\partial w_y}{\partial t}(x,y,t)&=-\beta _y(y)w_y(x,y,t)+(\beta _x(x)-\beta _y(y))\frac{\partial u}{\partial y}(x,y,t). \end{aligned}$$

  (45)

• 2. Using only first order spatial derivatives [13] (1SD)

  $$\begin{aligned} \frac{\partial u}{\partial t}(x,y,t)&=c^2(x,y)\bigg (\frac{\partial v_x}{\partial x}(x,y,t) +\frac{\partial v_y}{\partial y}(x,y,t) -w_x(x,y,t)\nonumber \\&\quad -w_y(x,y,t)\bigg )+\int \limits _{t_0}^tf(x,y,s)ds, \end{aligned}$$

  (46)
  $$\begin{aligned} \frac{\partial v_x}{\partial t}(x,y,t)&=-\beta _x(x)v_x(x,y,t)+\frac{\partial u}{\partial x}(x,y,t), \end{aligned}$$

  (47)
  $$\begin{aligned} \frac{\partial v_y}{\partial t}(x,y,t)&=-\beta _y(y)v_y(x,y,t)+\frac{\partial u}{\partial y}(x,y,t), \end{aligned}$$

  (48)
  $$\begin{aligned} \frac{\partial w_x}{\partial t}(x,y,t)&=\beta _x(x)\left( -w_x(x,y,t)+\frac{\partial v_x}{\partial x}(x,y,t)\right) , \end{aligned}$$

  (49)
  $$\begin{aligned} \frac{\partial w_y}{\partial t}(x,y,t)&=\beta _y(y)\left( -w_y(x,y,t)+\frac{\partial v_y}{\partial y}(x,y,t)\right) . \end{aligned}$$

  (50)

Two-dimensional elastic waves with PML, see Assi and Cobbold [4]:

$$\begin{aligned} \frac{\partial u_x}{\partial t}(x,y,t)&=v_x(x,y,t),\end{aligned}$$

(51)

$$\begin{aligned} \frac{\partial u_y}{\partial t}(x,y,t)&=v_y(x,y,t),\end{aligned}$$

(52)

$$\begin{aligned} \frac{\partial v_x}{\partial t}(x,y,t)&=-\big (\beta _x(x)+\beta _y(y)\big )v_x(x,y,t)-\beta _x(x)\beta _y(y)u_x(x,y,t)\nonumber \\&\quad +\frac{1}{\rho (x,y)}\left[ \frac{\partial }{\partial x}\left( T_{xx}(x,y,t)+w_{xx}(x,y,t)\right) \right. \nonumber \\&\quad \left. +\frac{\partial }{\partial y}\left( T_{xy}(x,y,t)+w_{xy}(x,y,t)\right) \right] +f_x(x,y,t),\end{aligned}$$

(53)

$$\begin{aligned} \frac{\partial v_y}{\partial t}(x,y,t)&=-\big (\beta _x(x)+\beta _y(y)\big )v_y(x,y,t)-\beta _x(x)\beta _y(y)u_y(x,y,t)\nonumber \\&\quad +\frac{1}{\rho }\left[ \frac{\partial }{\partial x}\left( T_{xy}(x,y,t)+w_{yx}(x,y,t)\right) \right. \nonumber \\&\quad \left. +\frac{\partial }{\partial y}\left( T_{yy}(x,y,t)+w_{yy}(x,y,t)\right) \right] +f_y(x,y,t),\end{aligned}$$

(54)

$$\begin{aligned} \frac{\partial T_{xx}}{\partial t}(x,y,t)&=\big (2\mu (x,y)+\lambda (x,y)\big )\frac{\partial v_x}{\partial x}(x,y,t)+\lambda (x,y)\frac{\partial v_y}{\partial y}(x,y,t),\end{aligned}$$

(55)

$$\begin{aligned} \frac{\partial T_{xy}}{\partial t}(x,y,t)&=\mu (x,y)\left( \frac{\partial v_x}{\partial y}(x,y,t)+\frac{\partial v_y}{\partial x}(x,y,t)\right) ,\end{aligned}$$

(56)

$$\begin{aligned} \frac{\partial T_{yy}}{\partial t}(x,y,t)&=\lambda (x,y)\frac{\partial v_x}{\partial x}(x,y,t)+\big (2\mu (x,y)+\lambda (x,y)\big )\frac{\partial v_y}{\partial y}(x,y,t),\end{aligned}$$

(57)

$$\begin{aligned} \frac{\partial w_{xx}}{\partial t}(x,y,t)&=-\beta _x(x)w_{xx}(x,y,t)+\big (\beta _y(y)-\beta _x(x)\big )\big (2\mu (x,y)+\lambda (x,y)\big )\frac{\partial u_x}{\partial x}(x,y,t),\end{aligned}$$

(58)

$$\begin{aligned} \frac{\partial w_{xy}}{\partial t}(x,y,t)&=-\beta _y(y)w_{xy}(x,y,t)+\big (\beta _x(x)-\beta _y(y)\big )\mu (x,y)\frac{\partial u_x}{\partial y}(x,y,t),\end{aligned}$$

(59)

$$\begin{aligned} \frac{\partial w_{yx}}{\partial t}(x,y,t)&=-\beta _x(x)w_{yx}(x,y,t)+\big (\beta _y(y)-\beta _x(x)\big )\mu (x,y)\frac{\partial u_y}{\partial x}(x,y,t),\end{aligned}$$

(60)

$$\begin{aligned} \frac{\partial w_{yy}}{\partial t}(x,y,t)&=-\beta _y(y)w_{yy}(x,y,t)+\big (\beta _x(x)-\beta _y(y)\big )\big (2\mu (x,y)+\lambda (x,y)\big )\frac{\partial u_y}{\partial y}(x,y,t), \end{aligned}$$

(61)

with the variables and parameters described in Table 2.

Table 2 Variables used in the equations and their description

The damping functions $\beta _z$, related to the absorption factor are defined as

$$\begin{aligned} \beta _z(z)=\left\{ \begin{array}{ll} 0,&{}\quad \text {if}\; d(z,\partial \varOmega )>\delta \\ \beta _0\left( \frac{d(z,\varOmega _1)}{\delta }\right) ^2,&{}\quad \text {if}\;d(z,\partial \varOmega )\le \delta \end{array}\right. ,\quad z=x,\;y \end{aligned}$$

(62)

where $d(z,\partial \varOmega )$ is the distance from z to the boundary of $\varOmega$, $\delta$ is the thickness of the PML domain, $\beta _0$ is the magnitude of the absorption factor, and $\varOmega _1$ is the numerical domain without the damping layer (physical domain). Thus, $\varOmega$ is composed by the union of $\varOmega _1$ and a damping layer of thickness $\delta$ extending on the boundary of $\varOmega _1$.

1.2 A.2: Discrete Framework

The spatial discretizations are based on a staggered grid using 4th and 8th order approximation of the spatial derivatives defined by Eqs. (20) and (21). Figures 15 and 16 describe the discrete space for the 1SD and 2SD formulation in 1D, and the 2SD and elastic formulations in 2D, respectively.

Fig. 15

Staggered grid in 1D with the relative positions of the (2SD) and (1SD) wave equation variables and parameters. $u,\;v$ and c are collocated (centered) and w is staggered in the grid

Fig. 16

Staggered grid in 2D with the relative positions of the (2SD and elastic) wave equations variables and parameters. $u,\;v,\;u_x,\;v_x,\;\rho$ and c are collocated

1.3 A.3: Numerical Benchmarks

We define the numerical experiments, called “Test Case” used through the paper. In all the tests we use a zero Dirichlet condition on the boundary of the domain $\varOmega$, a PML layer thickness of $\delta =0.8\,\text {km}$, a damping parameter $\beta _0=30$, and Ricker peak frequency of $f_0=25\,\text {Hz}$. If not otherwise stated, the initial condition for all the variables is zero. The particular benchmarks are then defined as follows:

Test Case #1: $\varOmega =[0,10.5\,\text {km}]$

$$\begin{aligned} c\equiv 1.524\,\text {km}/\text {s},\quad u_0(x)=((1-10(x-5.25)^2)e^{-10(x-5.25)^2},\quad f\equiv 0 \end{aligned}$$

Test Case #2: $\varOmega =[0,10.5\,\text {km}]$

$$\begin{aligned} c(x)&=\left\{ \begin{array}{ll} 1.524\;\text {km}/\text {s}, &{}\quad \text {if}\; x<5.25 \\ 3.048\;\text {km}/\text {s} &{}\quad \text {if}\; 5.25\le x< 7\\ 0.1524\;\text {km}/\text {s} &{}\quad \text {if}\; 7\le x \end{array}\right. ,\\ u_0&=\left\{ \begin{array}{ll} 0, &{}\quad \text {if}\; |x-2.6|\ge 0.01\\ e^{\frac{(x-2.6)^2}{(x-2.6)^2-0.01^2}}, &{}\quad \text {if}\; |x-2.6|<0.01 \end{array}\right. ,\quad f\equiv 0 \end{aligned}$$

Test Case #3: $\varOmega =[0,10.5\,\text {km}]$

$$\begin{aligned} c(x)&=\left\{ \begin{array}{ll} 1.524\;\text {km}/\text {s}, &{}\quad \text {if}\; x<5.25 \\ 3.048\;\text {km}/\text {s} &{}\quad \text {if}\; 5.25\le \\ \end{array}\right. ,\quad u_0\equiv 0,\\ f(x,t)&=\left\{ \begin{array}{ll} 0, &{}\quad \text {if}\; |x-2.6|\ge 0.01\\ e^{\frac{(x-2.6)^2}{(x-2.6)^2-0.01^2}}(1-f_0^2\pi ^2(t-t_0)^2)e^{-f_0^2\pi ^2(t-t_0)^2}, &{}\quad \text {if}\; |x-2.6|<0.01 \end{array}\right. \end{aligned}$$

Test Case #4: $\varOmega =[0,8\;\text {km}]\times [0,8\;\text {km}]$

$$\begin{aligned} c&\equiv 3\;\text {km}/\text {s},\quad u_0(x,y)=\left\{ \begin{array}{ll} 0,&{}\quad \text {if}\; \Vert (x,y)-(4,2)\Vert \ge 0.01 \\ e^{\frac{\Vert (x,y)-(4,2)\Vert ^2}{\Vert (x,y)-(4,2)\Vert ^2-0.01^2}},&{}\quad \text {if}\; \Vert (x,y)-(4,2)\Vert < 0.01 \end{array}\right. ,\\ f&\equiv 0 \end{aligned}$$

Test Case #5: $\varOmega =[0,8\;\text {km}]\times [0,8\;\text {km}]$

$$\begin{aligned} c(x,y)&=\left\{ \begin{array}{ll} 3\;\text {km}/\text {s}, &{}\quad \text {if}\; y\ge 4 \\ 6\;\text {km}/\text {s}, &{}\quad \text {if}\; (y<4\;\text {and}\; x\le 6)\;\text {or}\; (16/3< y<4\;\text {and}\; x>6)\\ 1\;\text {km}/\text {s}, &{}\quad \text {if}\; 6\le x\;\text {and}\; y\le 16/3 \end{array}\right. ,\\ u_0&=\left\{ \begin{array}{ll} 0,&{}\quad \text {if}\; \Vert (x,y)-(4,2)\Vert \ge 0.01 \\ e^{\frac{\Vert (x,y)-(4,2)\Vert ^2}{\Vert (x,y)-(4,2)\Vert ^2-0.01^2}},&{}\quad \text {if}\; \Vert (x,y)-(4,2)\Vert < 0.01 \end{array}\right. ,\quad f\equiv 0 \end{aligned}$$

Test Case #6: $\varOmega =[0,8\;\text {km}]\times [0,8\;\text {km}]$

$$\begin{aligned} c(x,y)&=\left\{ \begin{array}{ll} 3\;\text {km}/\text {s}, &{}\quad \text {if}\; y\ge 4 \\ 6\;\text {km}/\text {s}, &{}\quad \text {if}\; y<4\\ \end{array}\right. ,\quad u_0\equiv 0,\\ f(x,y,t)&=\left\{ \begin{array}{ll} 0, &{}\quad \text {if}\; \Vert (x,y)-(4,2)\Vert \ge 0.01\\ e^{\frac{\Vert (x,y)-(4,2)\Vert ^2}{\Vert (x,y)-(4,2)\Vert ^2-0.01^2}}(1-f_0^2\pi (t-t_0)^2)e^{-f_0^2\pi ^2(t-t_0)^2}, &{}\quad \text {if}\; \Vert (x,y)-(4,2)\Vert <0.01 \end{array}\right. \end{aligned}$$

Test Case #7: $\varOmega =[0,8\;\text {km}]\times [0,8\;\text {km}]$ for elastic waves

$$\begin{aligned} \rho&\equiv 0.25,\quad \mu (x,y)=\left\{ \begin{array}{ll} 1\;\text {km}/\text {s}, &{}\quad \text {if}\; y\ge 4 \\ 1.5\;\text {km}/\text {s}, &{}\quad \text {if}\; (y<4\;\text {and}\; x\le 6)\;\text {or}\; (16/3< y<4\; \text {and}\; x>6)\\ 2.25\;\text {km}/\text {s}, &{}\quad \text {if}\; 6\le x\; \text {and}\; y\le 16/3 \end{array}\right. \\ \lambda (x,y)&=\left\{ \begin{array}{ll} 8\;km/s, &{}\quad \text {if}\; y\ge 4 \\ 12\;km/s, &{}\quad \text {if}\;(y<4\; \text {and}\; x\le 6)\;\text {or}\; (16/3< y<4\;\text {and}\; x>6)\\ 18\;km/s, &{}\quad \text {if}\; 6\le x\;\text {and}\; y\le 16/3 \end{array}\right. ,\quad u_0\equiv 0,\\ f(x,y,t)&=\left\{ \begin{array}{ll} 0, &{}\quad \text {if}\; \Vert (x,y)-(4,2)\Vert \ge 0.01\\ e^{\frac{\Vert (x,y)-(4,2)\Vert ^2}{\Vert (x,y)-(4,2)\Vert ^2-0.01^2}}(1-f_0^2\pi (t-t_0)^2)e^{-f_0^2\pi ^2(t-t_0)^2}, &{}\quad \text {if}\; \Vert (x,y)-(4,2)\Vert <0.01 \end{array}\right. \end{aligned}$$

1.4 A.4: Spectrum Complementary Results

In this subsection, we present the spectral distribution results using the 2SD formulation, along with the real and imaginary limits of the spectrum for the elastic equations with PML.

Fig. 17

Eigenvalues on the complex plane (using coordinates $({\textit{Re}}(\lambda ), {\textit{Im}}(\lambda ))$) using a fourth-order spatial discretization of the acoustic wave equation in one dimension, with formulation 2SD, considering TC#3, for $\varDelta x=\{0.105,0.021,0.0105,0.0021\}$. For each $\varDelta x$, spectrum, the spectrum has a rectangular-shaped convex hull

Based on Fig. 17, it is evident that besides $\sigma ({\varvec{H}})$ being symmetric with respect to the real axis, the limits of the rectangle on the imaginary axis appear to have a linear relationship with $1/\varDelta x$. However, for the real part, the relationship is different, exhibiting a constant negative limit on the left side ($-\beta _0$) for the PML parameter $\beta _0>0$, and a small positive number on the right. In general, we observed similar results for other formulations (Fig. 18).

Fig. 18

Maximum imaginary parts (left) and lower bound of the real parts (right) of the eigenvalues of $\sigma ({\varvec{H}})$, for varying $1/\varDelta x$

For the acoustic 2SD equations, empirical studies, as the one shown in Fig. 6, indicate that the upper bound is a positive small number, smaller than 1. Since in all experiments this upper bound is always close to zero, this will not affect the estimates of the optimal ellipse for Faber polynomial approximations, as the ellipse size will be dominated by the imaginary axis bounds and the lower bound in the real axis. Therefore, a precise upper real bound will not be further required.

1.5 A.5: Stability and Dispersion

Here we present the operators and results of stability and dispersion for all the systems of equations considered in Section A.2 (assuming no PML and no source term), with a spatial discretization of fourth and eighth orders. At the end of the section are also presented the graphics of stability and dispersion of the elastic formulation (Figs. 19 and 20).

• 1. 1SD and 2SD in one dimension

  $$\begin{aligned} \varDelta t{\varvec{H}}=\frac{c\varDelta t}{\varDelta x}\begin{pmatrix} 0&{}g_{11}\\ g_{21}&{}0 \end{pmatrix},\quad \varDelta t{\varvec{H}}=\frac{c\varDelta t}{\varDelta x}\begin{pmatrix} 0&{}1\\ h_{11}&{}0 \end{pmatrix}. \end{aligned}$$

• 2. 1SD and 2SD in two dimension

  $$\begin{aligned} \varDelta t{\varvec{H}}=\frac{c\varDelta t}{\varDelta x}\begin{pmatrix} 0&{}g_{11}&{}g_{12}\\ g_{21}&{}0&{}0\\ g_{22}&{}0&{}0 \end{pmatrix},\quad \varDelta t{\varvec{H}}=\frac{c\varDelta t}{\varDelta x}\begin{pmatrix} 0&{}1\\ h_{11}+h_{22}&{}0 \end{pmatrix}. \end{aligned}$$

• 3. Elastic in two dimension (without considering the decoupled two first equations)

  $$\begin{aligned} \varDelta t{\varvec{H}}=\frac{\varDelta t}{\varDelta x}\frac{2\mu +\lambda }{\rho }\begin{pmatrix} 0&{}0&{}\frac{1}{2\mu +\lambda }g_{11}&{}\frac{1}{2\mu +\lambda }g_{12}&{}0\\ 0&{}0&{}0&{}\frac{1}{2\mu +\lambda }g_{21}&{}\frac{1}{2\mu +\lambda }g_{22}\\ \rho g_{21}&{}\frac{\rho \lambda }{2\mu +\lambda } g_{12}&{}0&{}0&{}0\\ \frac{\rho \mu }{2\mu +\lambda } g_{22}&{}\frac{\rho \mu }{2\mu +\lambda } g_{11}&{}0&{}0&{}0\\ \frac{\rho \lambda }{2\mu +\lambda } g_{21}&{}\rho g_{12}&{}0&{}0&{}0 \end{pmatrix}. \end{aligned}$$

Where

• 1. For 4th order

  $$\begin{aligned} g_{11}&=\frac{1}{24}\left( 27(1-e^{-ik_x\varDelta x})+e^{-2k_x\varDelta x}-e^{k_x\varDelta x}\right) \\ g_{12}&=\frac{1}{24}\left( 27(1-e^{-ik_y\varDelta x})+e^{-2k_y\varDelta x}-e^{k_y\varDelta x}\right) \\ g_{21}&=\frac{1}{24}\left( 27(e^{ik_x\varDelta x}-1)+e^{-k_x\varDelta x}-e^{2k_x\varDelta x}\right) \\ g_{22}&=\frac{1}{24}\left( 27(e^{ik_x\varDelta x}-1)+e^{-k_x\varDelta x}-e^{2k_x\varDelta x}\right) \\ h_{11}&=-\frac{1}{6}\cos (2\theta _x)+\frac{8}{3}\cos (\theta _x)-\frac{5}{2}\\ h_{22}&=-\frac{1}{6}\cos (2\theta _y)+\frac{8}{3}\cos (\theta _y)-\frac{5}{2} \end{aligned}$$

• 2. For 8th order

  $$\begin{aligned} g_{11}&=\frac{1225}{1024}\left( 1-e^{-ik_x\varDelta x}+\frac{1}{15}(e^{-2k_x\varDelta x}-e^{k_x\varDelta x})+\frac{1}{125}(e^{2k_x\varDelta x}-e^{-3k_x\varDelta x})\right. \\&\quad \left. +\frac{1}{1715}(e^{-4k_x\varDelta x}-e^{3k_x\varDelta x})\right) \\ g_{12}&=\frac{1225}{1024}\left( 1-e^{-ik_y\varDelta x}+\frac{1}{15}(e^{-2k_y\varDelta x}-e^{k_y\varDelta x})+\frac{1}{125}(e^{2k_y\varDelta x}-e^{-3k_y\varDelta x})\right. \\&\quad \left. +\frac{1}{1715}(e^{-4k_y\varDelta x}-e^{3k_y\varDelta x})\right) \\ g_{21}&=\frac{1225}{1024}\left( e^{ik_x\varDelta x}-1+\frac{1}{15}(e^{-k_x\varDelta x}-e^{2k_x\varDelta x})+\frac{1}{125}(e^{3k_x\varDelta x}-e^{-2k_x\varDelta x})\right. \\&\quad \left. +\frac{1}{1715}(e^{-3k_x\varDelta x}-e^{4k_x\varDelta x})\right) \\ g_{22}&=\frac{1225}{1024}\left( e^{ik_y\varDelta x}-1+\frac{1}{15}(e^{-k_y\varDelta x}-e^{2k_y\varDelta x})+\frac{1}{125}(e^{3k_y\varDelta x}-e^{-2k_y\varDelta x})\right. \\&\quad \left. +\frac{1}{1715}(e^{-3k_y\varDelta x}-e^{4k_y\varDelta x})\right) \\ h_{11}&=-\frac{1}{560}(e^{-4k_x\varDelta x}+e^{4k_x\varDelta x})+\frac{8}{315}(e^{-3k_x\varDelta x}+e^{3k_x\varDelta x})-\frac{1}{5}(e^{-2k_x\varDelta x}+e^{2k_x\varDelta x})\\&\quad +\frac{8}{5}(e^{-k_x\varDelta x}+e^{k_x\varDelta x})-\frac{205}{72}\\ h_{22}&=-\frac{1}{560}(e^{-4k_y\varDelta x}+e^{4k_y\varDelta x})+\frac{8}{315}(e^{-3k_y\varDelta x}+e^{3k_y\varDelta x})-\frac{1}{5}(e^{-2k_y\varDelta x}+e^{2k_y\varDelta x})\\&\quad +\frac{8}{5}(e^{-k_y\varDelta x}+e^{k_y\varDelta x})-\frac{205}{72} \end{aligned}$$

Fig. 19

Stability graphics of the Faber approximation method for different spatial discretizations, and polynomial degrees $m=\{3,4,\ldots ,40\}$, for the elastic formulation. The CFL number (left), and the measure of the operations number of Eq. (33) (right). Higher polynomial degrees implies in larger $c_{\text {CFL}}$, but this does not necessarily results in a decrease of the MVOs

Fig. 20

Dispersion studies of Faber approximation method using different spatial discretizations, dimensions, equations formulations, and polynomial degrees $m=\{3,4,\ldots ,40\}$. Higher polynomial degrees implies larger $\alpha _{R}$ and a non-monotonous decrease of the operations number

1.6 A.6: Convergence

In this section, we detail the construction of the RK(9-7) scheme employed for computing the reference solution. Furthermore, we provide the convergence and computational efficiency results achieved through the utilization of Faber polynomials in solving elastic wave propagation equations (Figs. 21 and 22).

RK(9-7) algorithm:

$$\begin{aligned} {\varvec{k}}_1&={\varvec{H}}{\varvec{u}}_n+{\varvec{f}}(t_n)\\ {\varvec{k}}_2&={\varvec{H}}\left( {\varvec{u}}_n+\frac{4}{63}\varDelta t{\varvec{k}}_1\right) +{\varvec{f}}\left( t_n+\frac{4}{63}\varDelta t\right) \\ {\varvec{k}}_3&={\varvec{H}}\left( {\varvec{u}}_n+\varDelta t\left[ \frac{1}{42}{\varvec{k}}_1+\frac{1}{14}{\varvec{k}}_2\right] \right) +{\varvec{f}}\left( t_n+\frac{2}{21}\varDelta t\right) \\ {\varvec{k}}_4&={\varvec{H}}\left( {\varvec{u}}_n+\varDelta t\left[ \frac{1}{28}{\varvec{k}}_1+\frac{3}{28}{\varvec{k}}_3\right] \right) +{\varvec{f}}\left( t_n+\frac{1}{7}\varDelta t\right) \\ {\varvec{k}}_5&={\varvec{H}}\left( {\varvec{u}}_n+\varDelta t\left[ \frac{12551}{19652}{\varvec{k}}_1-\frac{48363}{19652}{\varvec{k}}_3+\frac{10976}{4913}{\varvec{k}}_4\right] \right) +{\varvec{f}}\left( t_n+\frac{7}{17}\varDelta t\right) \\ {\varvec{k}}_6&={\varvec{H}}\left( {\varvec{u}}_n+\varDelta t\left[ -\frac{36616931}{27869184}{\varvec{k}}_1+\frac{2370277}{442368}{\varvec{k}}_3-\frac{255519173}{63700992}{\varvec{k}}_4+\frac{226798819}{445906944}{\varvec{k}}_5\right] \right) \\&\quad +{\varvec{f}}\left( t_n+\frac{13}{24}\varDelta t\right) \\ {\varvec{k}}_7&={\varvec{H}}\left( {\varvec{u}}_n+\varDelta t\left[ -\frac{10401401}{7164612}{\varvec{k}}_1+\frac{47383}{8748}{\varvec{k}}_3-\frac{4914455}{1318761}{\varvec{k}}_4-\frac{1498465}{7302393}{\varvec{k}}_5+\frac{2785280}{3739203}{\varvec{k}}_6\right] \right) \\&\quad +{\varvec{f}}\left( t_n+\frac{7}{9}\varDelta t\right) \\ {\varvec{k}}_8&={\varvec{H}}\left( {\varvec{u}}_n+\varDelta t\left[ \frac{181002080831}{17500000000}{\varvec{k}}_1-\frac{14827049601}{400000000}{\varvec{k}}_3+\frac{23296401527134463}{857600000000000}{\varvec{k}}_4\right. \right. \\&\quad +\left. \left. \frac{2937811552328081}{949760000000000}{\varvec{k}}_5-\frac{243874470411}{69355468750}{\varvec{k}}_6+\frac{2857867601589}{3200000000000}{\varvec{k}}_7\right] \right) \\&\quad +{\varvec{f}}\left( t_n+\frac{91}{100}\varDelta t\right) \\ {\varvec{k}}_9&={\varvec{H}}\left( {\varvec{u}}_n+\varDelta t\left[ -\frac{228380759}{19257212}{\varvec{k}}_1+\frac{4828803}{113948}{\varvec{k}}_3-\frac{331062132205}{10932626912}{\varvec{k}}_4\right. \right. \\&\left. \left. \quad -\frac{12727101935}{3720174304}{\varvec{k}}_5+\frac{22627205314560}{4940625496417}{\varvec{k}}_6-\frac{268403949}{461033608}{\varvec{k}}_7+\frac{3600000000000}{19176750553961}{\varvec{k}}_8\right] \right) \\&\quad +{\varvec{f}}\left( t_n+\varDelta t\right) \\ {\varvec{u}}_{n+1}&={\varvec{u}}_n+\varDelta t\left( \frac{95}{2366}{\varvec{k}}_1+\frac{3822231133}{16579123200}{\varvec{k}}_4+\frac{555164087}{2298419200}{\varvec{k}}_5+\frac{1279328256}{9538891505}{\varvec{k}}_6\right. \\&\quad +\left. \frac{5963949}{25894400}{\varvec{k}}_7+\frac{50000000000}{599799373173}{\varvec{k}}_8+\frac{28487}{712800}{\varvec{k}}_9\right) . \end{aligned}$$

Fig. 21

Approximation error of Faber polynomials using the elastic equations, to solve TC#7. The curves represent the error when using polynomial of degrees $m=\{5,10,\dots ,25\}$. Increasing the degree of the polynomial allows for larger steps in time, while keeping the approximation error smaller than a fixed threshold

Fig. 22

Convergence in polynomial order for the elastic equations, using different spatial discretization orders, and a wide range of polynomial degrees. The maximum $\varDelta t$ such that the error of Faber approximations is less than $10^{-6}$ is shown on the right. On the left, we show the number of operations using the values of $\varDelta t_{\text {max}}$ of the upper line. When the polynomial degree increases, the maximum allowed time-step size also increases, together with a decrease of the number of operations