Numerical Simulation of Cylindrical Solitary Waves in Periodic Media

Abstract

We study the behavior of nonlinear waves in a two-dimensional medium with density and stress relation that vary periodically in space. Efficient approximate Riemann solvers are developed for the corresponding variable-coefficient first-order hyperbolic system. We present direct numerical simulations of this multiscale problem, focused on the propagation of a single localized perturbation in media with strongly varying impedance. For the conditions studied, we find little evidence of shock formation. Instead, solutions consist primarily of solitary waves. These solitary waves are observed to be stable over long times and to interact in a manner approximately like solitons. The system considered has no dispersive terms; these solitary waves arise due to the material heterogeneity, which leads to strong reflections and effective dispersion.

Appendices

Appendix 1: Discretizations

1.1 Second-Order Clawpack Discretization

Clawpack is based on Lax–Wendroff discretization combined with TVD limiters, and is second order accurate in space and time [9]. The multidimensional Clawpack algorithms used here require the propagation of waves in both the normal and transverse directions at each cell edge. The Clawpack discretization takes the form

$$\begin{aligned} \mathbf{Q}_{ij}&= \mathbf{Q}_{ij}^{n}-\frac{\Delta t}{\Delta x}\left( \mathcal{A}^{+}\Delta \mathbf{Q}_{i-\frac{1}{2},j}^{n}+\mathcal{A}^{-}\Delta \mathbf{Q}_{i+\frac{1}{2},j}^{n}\right) \\&-\frac{\Delta t}{\Delta y}\left( \mathcal{B}^{+}\Delta \mathbf{Q}_{i,j-\frac{1}{2}}^{n}+\mathcal{B}^{-}\Delta \mathbf{Q}_{i,j+\frac{1}{2}}^{n}\right) \\&-\frac{\Delta t}{\Delta x}\left( \tilde{\varvec{F}}_{i+\frac{1}{2},j}^{n}-\tilde{\varvec{F}}_{i-\frac{1}{2},j}^{n}\right) -\frac{\Delta t}{\Delta y}\left( \tilde{\varvec{G}}_{i,j+\frac{1}{2}}^{n}-\tilde{\varvec{G}}_{i,j-\frac{1}{2}}^{n}\right) , \end{aligned}$$

where $\mathcal{A}^{\pm }\Delta \mathbf{Q}_{i\mp \frac{1}{2},j}^{n}$ and $\mathcal{B}^{\pm }\Delta \mathbf{Q}_{i,j\mp \frac{1}{2}}^{n}$ are first-order fluctuations computed by the normal Riemann solvers described in Sect. 5. The quantities $\tilde{\varvec{F}}_{i\pm \frac{1}{2},j}^{n}$ and $\tilde{\varvec{G}}_{i,j-\frac{1}{2}}^{n}$ are second-order corrections that include the fluctuations computed by the transverse Riemann solvers described in Sect. 5, as well as high-resolution approximations to $\mathbf{q}_{xx},\mathbf{q}_{yy}$. For details, see [11].

1.2 High-Order WENO Discretization Using SharpClaw

SharpClaw is based on the method of lines approach and uses fifth-order weighted essentially non-oscillatory (WENO) reconstruction in space with fourth-order strong stability preserving (SSP) Runge-Kutta integration in time [5]. SharpClaw requires propagation of waves only in the direction normal to each edge.

First, a WENO reconstruction of the solution is computed from the cell averages $Q_i$ to give high order accurate point values $q^L _{i-\frac{1}{2}}$ and $q^R _{i-\frac{1}{2}}$ just to the left and right (respectively) of each cell interfaces $x_{i-\frac{1}{2}}$. A Riemann solution is computed at each interface based on the reconstructed values there. The resulting fluctuations are used to update the adjacent cell averages. An additional term appears that is proportional to $\int _{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}Aq_x dx$. For conservative systems like (4), this term can be conveniently computed in terms of a fictitious internal Riemann problem in each cell [5]. The semi-discrete scheme takes the form

$$\begin{aligned} \frac{\partial \mathbf{Q}^{n+1}_{ij}}{\partial t}&= - \frac{1}{\Delta x}\left( \mathcal{{A^{-}}}\Delta {\mathbf{q}}_{{i+\frac{1}{2}},j} + \mathcal{{A^{+}}}\Delta {\mathbf{q}}_{{i-\frac{1}{2}},j} + \mathcal{{A}}\Delta {\mathbf{q}}_{ij}\right) \\&\qquad - \frac{1}{\Delta y}\left( \mathcal{{B^{-}}}\Delta {\mathbf{q}}_{i,{j+\frac{1}{2}}} + \mathcal{{B^{+}}}\Delta {\mathbf{q}}_{i,{j-\frac{1}{2}}} + \mathcal{{B}}\Delta {\mathbf{q}}_{ij}\right) \end{aligned}$$

For nonlinear systems with spatially-varying coefficients that are not piecewise-constant, the term $\mathcal{{A}}\Delta {\mathbf{q}}_{i}$ requires computation of certain integrals of the flux jacobian. This has not been implemented yet, so we use only Clawpack (not SharpClaw) when computing solutions in smoothly-varying media.

SharpClaw employs the same wave propagation Riemann solvers and user interface as Clawpack. In multi-dimensions SharpClaw requires propagation of waves only in the normal direction to each edge.

Appendix 2: Riemann Solvers

The Riemann solvers developed here follow the ideas of [10]. In particular, they are based on use of an all-shock solution and $f$-wave decomposition [2].

1.1 Normal Riemann Solvers

We assume that the density and bulk modulus are constant within each grid cell: $\rho =\rho _{ij}, K=K_{ij}$. In the case of smoothly varying media, this is an approximation. Then the system of conservation laws (4) can be written in the following quasilinear form within cell $i,j$:

$$\begin{aligned} \mathbf{q}_t + \mathbf{A}_{ij} \mathbf{q}_x + \mathbf{B}_{ij} \mathbf{q}_y = \mathbf{0}\quad \quad \forall (x,y) \in \left( x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}\right) \times \left( y_{j-\frac{1}{2}},y_{j+\frac{1}{2}}\right) \end{aligned}$$

(11)

where

$$\begin{aligned} \mathbf{q}= \begin{bmatrix} \epsilon \\ \rho u \\ \rho v \end{bmatrix}, \quad \mathbf{A}_{ij} = \mathbf{f}^{\prime }_{ij}(\mathbf{q})&= \begin{bmatrix} 0&-\frac{1}{\rho _{ij}}&0 \\ -\sigma _{\epsilon ,ij}&0&0 \\ 0&0&0 \end{bmatrix}, \quad \mathbf{B}_{ij} = \mathbf{g}^{\prime }_{ij}(\mathbf{q}) = \begin{bmatrix} 0&0&-\frac{1}{\rho _{ij}} \\ 0&0&0 \\ -\sigma _{\epsilon ,ij}&0&0 \end{bmatrix}.\nonumber \\ \end{aligned}$$

(12)

Note that $\sigma _{\epsilon ,ij}$ denotes the derivative of $\sigma (\epsilon ,x_i,y_j)$ with respect to $\epsilon$. For concreteness, we consider a Riemann problem in the x-direction, at interface $x_{{i-\frac{1}{2}},j}$ which consists of the hyperbolic system (4) with coefficients

$$\begin{aligned} \rho (x,y) = {\left\{ \begin{array}{ll} \rho _{i-1,j} &{} \forall x<x_{i-\frac{1}{2}}\\ \rho _{i,j} &{} \forall x>x_{i-\frac{1}{2}}\end{array}\right. }, \qquad K(x,y) = {\left\{ \begin{array}{ll} K_{i-1,j} &{} \forall x<x_{i-\frac{1}{2}}\\ K_{i,j} &{} \forall x>x_{i-\frac{1}{2}}; \end{array}\right. } \end{aligned}$$

(13)

and initial condition

$$\begin{aligned} \mathbf{q}(x,y)&= {\left\{ \begin{array}{ll} \mathbf{Q}_{i-1,j} &{} \forall x<x_{i-\frac{1}{2}}\\ \mathbf{Q}_{i,j} &{} \forall x>x_{i-\frac{1}{2}}\end{array}\right. }. \end{aligned}$$

(14)

In practice, $\mathbf{Q}_{ij}$ may represent a cell-average (in Clawpack) or a reconstructed value at the cell interface (in SharpClaw). The eigenvectors of $\mathbf{A}_{ij}(\mathbf{Q}_{ij})$ are

$$\begin{aligned} \mathbf{r}_{ij}^{1} = \begin{bmatrix} 1 \\ Z_{ij} \\ 0 \end{bmatrix}, \qquad \mathbf{r}_{ij}^{2} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \qquad \mathbf{r}_{ij}^{3} = \begin{bmatrix} -1 \\ Z_{ij} \\ 0 \end{bmatrix} \end{aligned}$$

with corresponding eigenvalues $\{-c_{ij},0,+c_{ij}\}$. Here $Z_{ij}=\sqrt{\rho _{ij}\sigma _{\epsilon ,ij}}$ is the impedance and $c_{ij}=\sqrt{\frac{\sigma _{\epsilon ,ij}}{\rho _{ij}}}$ is the sound speed. Here $\rho _{ij},Z_{ij}$ always represent cell averaged quantities.

In the linear case, each wave in the Riemann solution is a discontinuity proportional to the corresponding eigenvector in the material carrying the wave. Thus the solution can be found by decomposing the difference $\Delta {\mathbf{Q}}_{{i-\frac{1}{2}},j} = \mathbf{Q}_{ij}-\mathbf{Q}_{i-1,j}$ in terms of the following three eigenvectors:

$$\begin{aligned} \mathbf{r}_{{i-\frac{1}{2}},j}^{1} = \mathbf{r}_{i-1,j}^{1} = \begin{bmatrix} 1 \\ Z_{i-1,j} \\ 0 \end{bmatrix}, \quad \mathbf{r}_{{i-\frac{1}{2}},j}^{2} = \mathbf{r}^{2} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \quad \mathbf{r}_{{i-\frac{1}{2}},j}^{3} = \mathbf{r}_{i,j}^{3} = \begin{bmatrix} -1 \\ Z_{i,j} \\ 0 \end{bmatrix}\qquad \end{aligned}$$

(15)

In the nonlinear case, the Riemann solution also consists of three waves, one of which is a stationary shear wave with zero velocity. The other two waves may be rarefaction waves or shock waves, but the solution cannot include transonic rarefactions, since the $p$-system is derived in a Lagrangian frame. For reasons of computational efficiency, we will use an approximate all-shock solver. Shock waves in the Riemann solution correspond to traveling discontinuities that are proportional to $\mathbf{r}_{i-1,j}^1$ (left-going) or $\mathbf{r}_{i,j}^3$ (right-going). Equations for an exact all-shock Riemann solution can be derived in a straightforward manner by considering the Rankine-Hugoniot conditions. However, these lead to a coupled nonlinear system that is expensive to solve numerically. Instead, we again follow [10] and approximate the solution further by replacing the exact wave speeds with the sound speeds in each cell. The waves themselves are found following the $f$-wave approach [2], by decomposing the jump in the normal flux in terms of the eigenvectors (15):

$$\begin{aligned} \mathbf{F}_{i,j} - \mathbf{F}_{i-1,j}&= \beta _{{i-\frac{1}{2}},j}^{1}\mathbf{r}_{{i-\frac{1}{2}},j}^{1}+\beta _{{i-\frac{1}{2}},j}^{2}\mathbf{r}_{{i-\frac{1}{2}},j}^{2}+\beta _{{i-\frac{1}{2}},j}^{3}\mathbf{r}_{{i-\frac{1}{2}},j}^{3} \\&= \mathbf{\mathcal {Z}}_{{i-\frac{1}{2}},j}^{1}+\mathbf{\mathcal {Z}}_{{i-\frac{1}{2}},j}^{2}+\mathbf{\mathcal {Z}}_{{i-\frac{1}{2}},j}^{3}, \end{aligned}$$

where $\mathbf{F}_{ij}=\mathbf{f}(\mathbf{Q}_{ij})$. The second wave is not needed in the numerical solution, since it has velocity zero and thus does not affect the solution value in either cell. The other two waves are accumulated into left- and right-going fluctuations, which are quantities that consider the net effect of all left- and right-going waves respectively:

$$\begin{aligned} \mathcal{{A^{-}}}\Delta {\mathbf{Q}}_{{i-\frac{1}{2}},j}&= \mathbf{\mathcal {Z}}_{{i-\frac{1}{2}},j}^1 \end{aligned}$$

(16)

$$\begin{aligned} \mathcal{{A^{+}}}\Delta {\mathbf{Q}}_{{i-\frac{1}{2}},j}&= \mathbf{\mathcal {Z}}_{{i-\frac{1}{2}},j}^3. \end{aligned}$$

(17)

The normal Riemann solver for the y-direction uses the same approach. To solve the Riemann problem at $(x_i,y_{j-\frac{1}{2}})$, in place of (15) we use the eigenvectors

$$\begin{aligned} \mathbf{r}_{i,{j-\frac{1}{2}}}^{1} = \mathbf{r}_{i,j-1}^{1} = \begin{bmatrix} 1 \\ 0 \\ Z_{i,j-1} \end{bmatrix}, \quad \mathbf{r}_{i,{j-\frac{1}{2}}}^{2} = \mathbf{r}^{2} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad \mathbf{r}_{i,{j-\frac{1}{2}}}^{3} = \mathbf{r}_{i,j}^{3} = \begin{bmatrix} -1 \\ 0 \\ Z_{i,j} \end{bmatrix}.\qquad \end{aligned}$$

(18)

The normal flux difference is decomposed in terms of these eigenvectors:

$$\begin{aligned} \mathbf{G}_{i,j} - \mathbf{G}_{i,j-1}&= \beta _{i,{j-\frac{1}{2}}}^{1}\mathbf{r}_{i,{j-\frac{1}{2}}}^{1}+\beta _{i,{j-\frac{1}{2}}}^{2}\mathbf{r}_{i,{j-\frac{1}{2}}}^{2}+\beta _{i,{j-\frac{1}{2}}}^{3}\mathbf{r}_{i,{j-\frac{1}{2}}}^{3} \\&= \mathbf{\mathcal {Z}}_{i,{j-\frac{1}{2}}}^{1}+\mathbf{\mathcal {Z}}_{i,{j-\frac{1}{2}}}^{2}+\mathbf{\mathcal {Z}}_{i,{j-\frac{1}{2}}}^{3}, \end{aligned}$$

where $\mathbf{G}_{ij}=\mathbf{g}(Q_{ij})$. Finally, the waves are accumulated into up- and down-going fluctuations:

$$\begin{aligned} \mathcal{{B^{-}}}\Delta {\mathbf{Q}}_{i,{j-\frac{1}{2}}}&= \mathbf{\mathcal {Z}}_{i,{j-\frac{1}{2}}}^1 \end{aligned}$$

(19)

$$\begin{aligned} \mathcal{{B^{+}}}\Delta {\mathbf{Q}}_{i,{j-\frac{1}{2}}}&= \mathbf{\mathcal {Z}}_{i,{j-\frac{1}{2}}}^3. \end{aligned}$$

(20)

As observed in [10] for the 1D $p$-system, this approach leads to efficient approximate Riemann solvers that are conservative and provide good accuracy at least for weakly nonlinear problems.

1.2 Transverse Riemann Solvers

The multidimensional Clawpack algorithm makes use of a transverse Riemann solver that decomposes the horizontally traveling waves into up- and down-going corrections and the vertical traveling waves into right- and left-going corrections. These second-order corrections capture the effect of corner transport.

Transverse corrections to the vertical fluctuations are computed by decomposing the horizontal fluctuations into up- and down-going parts using the eigenvectors (18):

$$\begin{aligned} \mathcal A ^{\pm }\Delta \mathbf{Q}_{i-\frac{1}{2},j} = \gamma _{i,j-\frac{1}{2}}^{1}\varvec{r}_{i,j-\frac{1}{2}}^{1}+\gamma _{i,j-\frac{1}{2}}^{2}\varvec{r}_{i,j-\frac{1}{2}}^{2}+\gamma _{i,j-\frac{1}{2}}^{3}\varvec{r}_{i,j-\frac{1}{2}}^{3}. \end{aligned}$$

(21)

The speed of the waves in the vertical direction determines the amount of horizontal fluctuation that must be added to the vertical one. These speeds are given by the eigenvalues of $\mathbf{B}_{i,{j-\frac{1}{2}}}$, which are $s_{i,j-\frac{1}{2}}^{1}=-c_{i,j-\frac{1}{2}}$, $s_{i,j-\frac{1}{2}}^{2}=0$ and $s_{i,j-\frac{1}{2}}^{3}=c_{i,j-\frac{1}{2}}$. Finally, the corrections to the vertical fluctuations are:

$$\begin{aligned} \mathcal B ^{-}\mathcal A ^{\pm }\Delta \mathbf{Q}_{i-\frac{1}{2},j}&= s_{i,j-\frac{1}{2}}^{1}\gamma _{i,j-\frac{1}{2}}^{1}\varvec{r}_{i,j-\frac{1}{2}}^{1},\\ \mathcal B ^{+}\mathcal A ^{\pm }\Delta \mathbf{Q}_{i-\frac{1}{2},j}&= s_{i,j-\frac{1}{2}}^{3}\gamma _{i,j-\frac{1}{2}}^{3}\varvec{r}_{i,j-\frac{1}{2}}^{3}. \end{aligned}$$

The transverse corrections to the horizontal fluctuations are obtained in an analogous way. Those corrections are:

$$\begin{aligned} \mathcal A ^{-}\mathcal B ^{\pm }\Delta \mathbf{Q}_{i,j-\frac{1}{2}}&= s_{i-\frac{1}{2},j}^{1}\gamma _{i-\frac{1}{2},j}^{1}\varvec{r}_{i-\frac{1}{2},j}^{1},\\ \mathcal A ^{+}\mathcal B ^{\pm }\Delta \mathbf{Q}_{i,j-\frac{1}{2}}&= s_{i-\frac{1}{2},j}^{3}\gamma _{i-\frac{1}{2},j}^{3}\varvec{r}_{i-\frac{1}{2},j}^{3}. \end{aligned}$$