Particle-without-Particle: A Practical Pseudospectral Collocation Method for Linear Partial Differential Equations with Distributional Sources

Abstract

Partial differential equations with distributional sources—in particular, involving (derivatives of) delta distributions—have become increasingly ubiquitous in numerous areas of physics and applied mathematics. It is often of considerable interest to obtain numerical solutions for such equations, but any singular (“particle”-like) source modeling invariably introduces nontrivial computational obstacles. A common method to circumvent these is through some form of delta function approximation procedure on the computational grid; however, this often carries significant limitations on the efficiency of the numerical convergence rates, or sometimes even the resolvability of the problem at all. In this paper, we present an alternative technique for tackling such equations which avoids the singular behavior entirely: the “Particle-without-Particle” method. Previously introduced in the context of the self-force problem in gravitational physics, the idea is to discretize the computational domain into two (or more) disjoint pseudospectral (Chebyshev–Lobatto) grids such that the “particle” is always at the interface between them; thus, one only needs to solve homogeneous equations in each domain, with the source effectively replaced by jump (boundary) conditions thereon. We prove here that this method yields solutions to any linear PDE the source of which is any linear combination of delta distributions and derivatives thereof supported on a one-dimensional subspace of the problem domain. We then implement it to numerically solve a variety of relevant PDEs: hyperbolic (with applications to neuroscience and acoustics), parabolic (with applications to finance), and elliptic. We generically obtain improved convergence rates relative to typical past implementations relying on delta function approximations.

Appendices

Proof of Distributional Identity

The base case ($n=0$) reads $f(x,{\mathbf {y}})\delta (x-x_{p})\equiv f_{p}\delta (x-x_{p})$. It is trivial to see that this holds.

Now assume the identity holds for $n=k$. Then we must prove that it holds for $n=k+1$. In other words, we wish to show that:

$$\begin{aligned} \left\langle f\delta _{\left( p\right) }^{\left( k+1\right) },\phi \right\rangle =\left\langle \left( -1\right) ^{n}\sum _{j=0}^{n}\left( -1\right) ^{j}\left( {\begin{array}{c}n\\ j\end{array}}\right) f_{p}^{\left( n-j\right) }\delta _{\left( p\right) }^{\left( j\right) },\phi \right\rangle . \end{aligned}$$

(84)

We begin with the LHS, and compute:

$$\begin{aligned} \left\langle f\delta _{\left( p\right) }^{\left( k+1\right) },\phi \right\rangle&= \int _{{\mathscr {I}}}\mathrm{d}x\,f\left( x,{\mathbf {y}}\right) \delta _{\left( p\right) }^{\left( k+1\right) }\left( x\right) \phi \left( x\right) \end{aligned}$$

(85)

$$\begin{aligned}&= -\int _{{\mathscr {I}}}\mathrm{d}x\,\left( \phi '\left( x\right) f\left( x,{\mathbf {y}}\right) +\phi \left( x\right) f^{'}\left( x,{\mathbf {y}}\right) \right) \delta _{\left( p\right) }^{\left( k\right) }\left( x\right) , \end{aligned}$$

(86)

using integration by parts. Now, inserting the induction hypothesis,

$$\begin{aligned} \left\langle f\delta _{\left( p\right) }^{\left( k+1\right) },\phi \right\rangle&= -\int _{{\mathscr {I}}}\mathrm{d}x\,\bigg \{\phi '\left( x\right) \left( -1\right) ^{k}\sum _{j=0}^{k}\left( -1\right) ^{j}\left( {\begin{array}{c}k\\ j\end{array}}\right) f_{p}^{\left( k-j\right) }\delta _{\left( p\right) }^{\left( j\right) }\left( x\right) \nonumber \\&\quad +\phi \left( x\right) \left( -1\right) ^{k}\sum _{l=0}^{k}\left( -1\right) ^{l}\left( {\begin{array}{c}k\\ l\end{array}}\right) f_{p}^{\left( k+1-l\right) }\delta _{\left( p\right) }^{\left( j\right) }\left( x\right) \bigg \} \end{aligned}$$

(87)

$$\begin{aligned}&= \left( -1\right) ^{k+1}\bigg \{\sum _{j=0}^{k}\left( -1\right) ^{j}\left( {\begin{array}{c}k\\ j\end{array}}\right) f_{p}^{\left( k-j\right) }\int _{{\mathscr {I}}}\mathrm{d}x\,\phi '\left( x\right) \delta _{\left( p\right) }^{\left( j\right) }\left( x\right) \nonumber \\&\quad +\sum _{l=0}^{k}\left( -1\right) ^{l}\left( {\begin{array}{c}k\\ l\end{array}}\right) f_{p}^{\left( k+1-l\right) }\int _{{\mathscr {I}}}\mathrm{d}x\,\phi \left( x\right) \delta _{\left( p\right) }^{\left( l\right) }\left( x\right) \bigg \} \end{aligned}$$

(88)

Observe that, via integration by parts, $\langle \delta _{\left( p\right) }^{\left( j\right) },\phi \rangle =(-1)^{j}\phi _{p}^{\left( j\right) }$. Hence, the above simplifies to:

$$\begin{aligned} \left\langle f\delta _{\left( p\right) }^{\left( k+1\right) },\phi \right\rangle&= \left( -1\right) ^{k+1}\bigg \{\sum _{j=0}^{k}\left( {\begin{array}{c}k\\ j\end{array}}\right) f_{p}^{\left( k-j\right) }\phi _{p}^{\left( j+1\right) }+\sum _{l=0}^{k}\left( {\begin{array}{c}k\\ l\end{array}}\right) f_{p}^{\left( k+1-l\right) }\phi _{p}^{\left( l\right) }\bigg \} \end{aligned}$$

(89)

$$\begin{aligned}&= \left( -1\right) ^{k+1}\bigg \{\sum _{j=1}^{k+1}\left( {\begin{array}{c}k\\ j-1\end{array}}\right) f_{p}^{\left( k+1-j\right) }\phi _{p}^{\left( j\right) }+\sum _{l=0}^{k}\left( {\begin{array}{c}k\\ l\end{array}}\right) f_{p}^{\left( k+1-l\right) }\phi _{p}^{\left( l\right) }\bigg \} \end{aligned}$$

(90)

$$\begin{aligned}&= \left( -1\right) ^{k+1}\bigg \{ f_{p}^{\left( k+1\right) }\phi _{p}+\sum _{j=1}^{k+1}\left[ \left( {\begin{array}{c}k\\ j-1\end{array}}\right) +\left( {\begin{array}{c}k\\ j\end{array}}\right) \right] f_{p}^{\left( k+1-j\right) }\phi _{p}^{\left( j\right) }+f_{p}\phi _{p}^{\left( k+1\right) }\bigg \} \end{aligned}$$

(91)

after rearranging the sum terms. Using the recursive formula for the binomial coefficient, $\left( {\begin{array}{c}k\\ j-1\end{array}}\right) +\left( {\begin{array}{c}k\\ j\end{array}}\right) =\left( {\begin{array}{c}k+1\\ j\end{array}}\right)$, and then including the first and last terms in (91) into the sum, we get:

$$\begin{aligned} \left\langle f\delta _{\left( p\right) }^{\left( k+1\right) },\phi \right\rangle =\left( -1\right) ^{k+1}\sum _{j=0}^{k+1}\left( {\begin{array}{c}k+1\\ j\end{array}}\right) f_{p}^{\left( k+1-j\right) }\phi _{p}^{\left( j\right) }. \end{aligned}$$

(92)

Now using $\phi _{p}^{\left( j\right) }=\langle \delta _{\left( p\right) },\phi ^{\left( j\right) }\rangle =(-1)^{j}\langle \delta _{\left( p\right) }^{\left( j\right) },\phi \rangle$, we finally obtain

$$\begin{aligned} \left\langle f\delta _{\left( p\right) }^{\left( k+1\right) },\phi \right\rangle&= \int _{{\mathscr {I}}}\mathrm{d}x\,\phi \left( x\right) \bigg [\left( -1\right) ^{k+1}\sum _{j=0}^{k+1}\left( -1\right) ^{j}\left( {\begin{array}{c}k+1\\ j\end{array}}\right) f_{p}^{\left( k+1-j\right) }\delta _{\left( p\right) }^{\left( j\right) }\left( x\right) \bigg ] \end{aligned}$$

(93)

$$\begin{aligned}&= \left\langle \left( -1\right) ^{k+1}\sum _{j=0}^{k+1}\left( -1\right) ^{j}\left( {\begin{array}{c}k+1\\ j\end{array}}\right) f_{p}^{\left( k+1-j\right) }\delta _{\left( p\right) }^{\left( j\right) },\phi \right\rangle , \end{aligned}$$

(94)

which is what we wanted to prove.

Pseudospectral Collocation Methods

We use this appendix to describe very cursorily the PSC methods used for the numerical schemes in this paper and to introduce some notation in relation thereto. For good detailed expositions see, for example, Refs. [68, 75, 76].

We work on Chebyshev–Lobatto (CL) computational grids. On any domain $[a,b]={\mathscr {D}}\subseteq {\mathscr {I}}$, these comprise the (non-uniformly spaced) set of N points $\{X_{i}\}_{i=0}^{N}\subset {\mathscr {D}}$ obtained by projecting onto ${\mathscr {D}}$ those points located at equal angles on a hypothetical semicircle having ${\mathscr {D}}$ as its diameter. That is to say, the CL grid on the “standard” spectral domain ${\mathscr {D}}^{\text {s}}=[-1,1]$ is given by

$$\begin{aligned} X_{i}^{\text {s}}=-\cos \left( \frac{\pi i}{N}\right) \,,\quad \forall 0\le i\le N\,, \end{aligned}$$

(95)

which can straightforwardly be transformed (by shifting and stretching) to the desired grid on ${\mathscr {D}}$. For any function $f:{\mathscr {D}}\rightarrow {\mathbb {R}}$ we denote via a subscript its value at the i-th CL point, $f(X_{i})=f_{i}$, and in slanted boldface the vector containing all such values,

$$\begin{aligned} {\varvec{f}}=\left[ \begin{array}{c} f_{0}\\ f_{1}\\ \vdots \\ f_{N} \end{array}\right] \,. \end{aligned}$$

(96)

There exists an $(N+1)\times (N+1)$ matrix ${\mathbb {D}}$, the so-called CL differentiation matrix, such that the derivative values of f can be approximated simply by applying it to (96), i.e. ${\varvec{f}}'={\mathbb {D}}{\varvec{f}}$. For convenience, we also employ the notation ${\mathbb {M}}(r_{\text {i}}:r_{\text {f}},c_{\text {i}}:c_{\text {f}})$ to refer to the part of any matrix ${\mathbb {M}}$ from the $r_{\text {i}}$-th to the $r_{\text {f}}$-th row and from the $c_{\text {i}}$-th to the $c_{\text {f}}$-th column. (A simple “ : ” indicates taking all rows/columns.)

Numerical Schemes for Distributionally-Sourced PDEs

1.1 First-Order Hyperbolic PDEs

We apply a first order in time finite difference scheme to the homogeneous PDEs; thus, prior to imposing BCs/JCs, the equations become $\frac{1}{\Delta t}({\varvec{u}}_{k+1}^{\pm }-{\varvec{u}}_{k}^{\pm })=-{\mathbb {D}}^{\pm }{\varvec{u}}_{k}^{\pm }$, where the vectors ${\varvec{u}}_{k}^{\pm }$ contain the values of the solutions on the CL grids at the k-th time step, ${\mathbb {D}}^{\pm }$ is the CL differentiation matrix on the respective domains, and $\Delta t$ is our time step. We can rewrite the discretized PDE as ${\varvec{u}}_{k+1}^{\pm }={\varvec{u}}_{k}^{\pm }-\Delta t{\mathbb {D}}^{\pm }{\varvec{u}}_{k}^{\pm }={\varvec{s}}_{k}^{\pm }$. To impose the BC and JC, we modify the equations as follows:

$$\begin{aligned} \left[ \begin{array}{c} {\varvec{u}}_{k+1}^{-}\\ \hline {\varvec{u}}_{k+1}^{+} \end{array}\right] =\left[ \begin{array}{c} u_{N,k}^{+}\\ {\varvec{s}}_{k}^{-}(2:N+1)\\ \hline u_{N,k}^{-}+g_{k}\\ {\varvec{s}}_{k}^{+}(2:N+1) \end{array}\right] . \end{aligned}$$

(97)

Similarly, for our neuroscience application, we discretize the PDE using a first order finite difference scheme: $\frac{1}{\Delta t}({\varvec{\rho }}_{k+1}^{\pm }-{\varvec{\rho }}_{k}^{\pm })=-{\mathbb {D}}^{\pm }{\varvec{R}}_{k}^{\pm }$ where $R_{i,k}^{\pm }=(1-V_{i}^{\pm })\rho _{i,k}^{\pm }$. Hence, prior to imposing the BC/JC, we have ${\varvec{\rho }}_{k+1}^{\pm }={\varvec{\rho }}_{k}^{\pm }-\Delta t{\mathbb {D}}^{\pm }{\varvec{R}}_{k}^{\pm }={\varvec{s}}_{k}^{\pm }$. To impose the BC/JC, we just modify the equations accordingly:

$$\begin{aligned} \left[ \begin{array}{c} {\varvec{\rho }}_{k+1}^{-}\\ \hline {\varvec{\rho }}_{k+1}^{+} \end{array}\right] =\left[ \begin{array}{c} 0\\ {\varvec{s}}_{k}^{-}(2:N+1)\\ \hline \rho _{N,k}^{-}+\frac{1-L}{1-V_{*}}\rho _{N,k}^{+}\\ {\varvec{s}}_{k}^{+}(2:N+1) \end{array}\right] . \end{aligned}$$

(98)

1.2 Parabolic PDEs

In these problems, we have moving boundaries for the CL grids (since the location of the singular source is time-dependent). The mapping for transforming the standard (fixed) spectral domain $[-1,1]$ into an arbitrary (time-dependent) one, say ${\mathscr {D}}=[a(t),b(t)]$, is given by

$$\begin{aligned} {\mathscr {V}}\times \left[ 0,1\right] \rightarrow \,&{\mathscr {V}}\times {\mathscr {D}} \end{aligned}$$

(99)

$$\begin{aligned} \left( T,X\right) \mapsto \,&\left( t\left( T\right) ,x\left( T,X\right) \right) \,, \end{aligned}$$

(100)

where

$$\begin{aligned} t\left( T\right)&= T\,, \end{aligned}$$

(101)

$$\begin{aligned} x\left( T,X\right)&= \frac{b-a}{2}X+\frac{a+b}{2}\,. \end{aligned}$$

(102)

For transforming back, we have

$$\begin{aligned} {\mathscr {V}}\times {\mathscr {D}}\rightarrow \,&{\mathscr {V}}\times \left[ 0,1\right] \end{aligned}$$

(103)

$$\begin{aligned} \left( t,x\right) \mapsto \,&\left( T\left( t\right) ,X\left( t,x\right) \right) \,, \end{aligned}$$

(104)

where

$$\begin{aligned} T\left( t\right)&= t\,, \end{aligned}$$

(105)

$$\begin{aligned} X\left( t,x\right)&= \frac{2x-a-b}{b-a}\,. \end{aligned}$$

(106)

Thus, for any function f(t, x) in these problems, we must take care to express the time partial using the chain rule as

$$\begin{aligned} \frac{\partial f}{\partial t}&= \frac{\partial f}{\partial T}\frac{\partial T}{\partial t}+\frac{\partial f}{\partial X}\frac{\partial X}{\partial t} \end{aligned}$$

(107)

$$\begin{aligned}&= \frac{\partial f}{\partial T}-\frac{2}{\left( b-a\right) ^{2}}\left[ \left( b-x\right) {\dot{a}}+\left( x-a\right) {\dot{b}}\right] \frac{\partial f}{\partial X}\,, \end{aligned}$$

(108)

where in the second line we have used (105)–(106).

Now, let us use this to formulate the numerical schemes for our problems—first, for the heat equation. Let ${\mathbb {D}}_{k}^{\pm }$ denote the CL differentiation matrices on each of the two domains at the k-th time step. Then, using (108), we have here the following finite difference formula for the homogeneous PDEs prior to imposing BCs/JCs: $\frac{1}{\Delta t}({\varvec{u}}_{k+1}^{\pm }-{\varvec{u}}_{k}^{\pm })=({\mathbb {D}}_{k}^{\pm })^{2}{\varvec{u}}_{k}^{\pm }-{\mathbb {C}}_{k}^{\pm }{\mathbb {D}}{\varvec{u}}_{k}^{\pm }$, where ${\mathbb {D}}$ is the CL differentiation matrix on $[-1,1]$ and ${\mathbb {C}}_{k}^{-}=\mathrm{diag}([2/(x_{p}(t_{k}))^{2}][(-x_{i}^{-}){\dot{x}}_{p}(t_{k})])$, ${\mathbb {C}}_{k}^{+}=\mathrm{diag}([2/(1-x_{p}(t_{k}))^{2}][(x_{i}^{+}-1){\dot{x}}_{p}(t_{k})])$. Thus ${\varvec{u}}_{k+1}^{\pm }={\varvec{u}}_{k}^{\pm }+\Delta t[({\mathbb {D}}_{k}^{\pm })^{2}-{\mathbb {C}}_{k}^{\pm }{\mathbb {D}}]{\varvec{u}}_{k}^{\pm }={\varvec{s}}_{k}^{\pm }$. We can implement the BCs and JCs, by modifying the first and last equations on each domain:

$$\begin{aligned} \left[ \begin{array}{cccc|cccc} 1 &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} 1 &{} \cdots &{} 0 &{} &{} 0\\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} &{} &{} \ddots \\ 0 &{} 0 &{} \cdots &{} 1 &{} &{} &{} &{} 0\\ \hline 0 &{} &{} &{} &{} &{} &{} {\mathbb {D}}_{k}^{+}(1,:)\\ &{} 0 &{} &{} &{} 0 &{} 1 &{} \cdots &{} 0\\ &{} &{} \ddots &{} &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ &{} &{} &{} 0 &{} 0 &{} 0 &{} \cdots &{} 1 \end{array}\right] \left[ \begin{array}{c} \\ {\varvec{u}}_{k+1}^{-}\\ \\ \hline \\ {\varvec{u}}_{k+1}^{+}\\ \\ \end{array}\right] =\left[ \begin{array}{c} 0\\ {\varvec{s}}_{k}^{-}(2:N)\\ u_{0,k}^{+}\\ \hline {\mathbb {D}}_{k}^{-}(N,:){\varvec{u}}_{k}^{-}-\lambda \\ {\varvec{s}}_{k}^{+}(2:N)\\ 0 \end{array}\right] . \end{aligned}$$

(109)

Note that we are actually introducing an error by using (for convenience and ease of adaptability) ${\mathbb {D}}_{k}^{+}$ instead of ${\mathbb {D}}_{k+1}^{+}$ on the LHS (in the equation for $u_{0,k+1}^{+}$). However, one can easily convince oneself that ${\mathbb {D}}_{k+1}^{+}-{\mathbb {D}}_{k}^{+}={\mathcal {O}}(\Delta t)$, which is already the order of the error of the finite difference scheme, so we are not actually introducing any new error in this way. Furthermore, because we use up the last equation for ${\varvec{u}}_{k}^{-}$ to impose the JC on u (i.e. we do not have an equation for $u_{N,k}^{-}$), we must use the derivative at the previous point (i.e., at $u_{N-1,k}^{-}$) in order to impose the derivative JC. Hence on the RHS, we use ${\mathbb {D}}_{k}^{-}(N,:)$ instead of ${\mathbb {D}}_{k}^{-}(N+1,:)$.

The scheme for the finance model is analogous. We use again the first-order finite-difference method for the homogeneous equations, $\frac{1}{\Delta t}({\varvec{f}}_{k+1}^{\sigma }-{\varvec{f}}_{k}^{\sigma })=({\mathbb {D}}_{k}^{\sigma })^{2}{\varvec{f}}_{k}^{\sigma }-{\mathbb {C}}_{k}^{\sigma }{\mathbb {D}}{\varvec{f}}_{k}^{\sigma }$ with the matrices ${\mathbb {C}}_{k}^{\sigma }$ defined similarly to those in the heat equation problem (again using (108)); thus ${\varvec{f}}_{k+1}^{\sigma }={\varvec{f}}_{k}^{\sigma }+\Delta t[({\mathbb {D}}_{k}^{\sigma })^{2}-{\mathbb {C}}_{k}^{\sigma }{\mathbb {D}}]{\varvec{f}}_{k}^{\sigma }={\varvec{s}}_{k}^{\sigma }$. To impose the BCs/JCs, we modify the equations appropriately:

$$\begin{aligned} \left[ \begin{array}{ccccc} &{} &{} {\mathbb {D}}_{k}^{-}(1,:)\\ 0 &{} 1 &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 1 &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} 1 \end{array}\right] \left[ \begin{array}{c} \\ \\ {\varvec{f}}_{k+1}^{-}\\ \\ \\ \end{array}\right]&=\left[ \begin{array}{c} 0\\ s_{1,k}^{-}\\ \vdots \\ s_{N,k}^{-}\\ f_{0,k}^{0} \end{array}\right] , \end{aligned}$$

(110)

$$\begin{aligned} \left[ \begin{array}{ccccc} &{} &{} {\mathbb {D}}_{k}^{0}(1,:)\\ 0 &{} 1 &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 1 &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} 1 \end{array}\right] \left[ \begin{array}{c} \\ \\ {\varvec{f}}_{k+1}^{0}\\ \\ \\ \end{array}\right]&=\left[ \begin{array}{c} {\mathbb {D}}_{k}^{-}(N,:){\varvec{f}}_{k}^{-}-\lambda _{k}\\ s_{1,k}^{0}\\ \vdots \\ s_{N,k}^{0}\\ f_{0,k}^{+} \end{array}\right] , \end{aligned}$$

(111)

$$\begin{aligned} \left[ \begin{array}{ccccc} &{} &{} {\mathbb {D}}_{k}^{+}(1,:)\\ 0 &{} 1 &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 1 &{} 0\\ &{} &{} {\mathbb {D}}_{k}^{+}(N+1,:) \end{array}\right] \left[ \begin{array}{c} \\ \\ {\varvec{f}}_{k+1}^{+}\\ \\ \\ \end{array}\right]&=\left[ \begin{array}{c} {\mathbb {D}}_{k}^{0}(N,:){\varvec{f}}_{k}^{0}+\lambda _{k}\\ s_{1,k}^{+}\\ \vdots \\ s_{N,k}^{+}\\ 0 \end{array}\right] . \end{aligned}$$

(112)

1.3 Second-Order Hyperbolic PDEs

We again apply a first order in time finite difference scheme to the homogeneous PDEs; prior to imposing BCs/JCs, the equations become

$$\begin{aligned} \frac{1}{\Delta t}\left( \left[ \begin{array}{c} {\varvec{u}}_{k+1}^{\pm }\\ {\varvec{v}}_{k+1}^{\pm }\\ {\varvec{w}}_{k+1}^{\pm } \end{array}\right] -\left[ \begin{array}{c} {\varvec{u}}_{k}^{\pm }\\ {\varvec{v}}_{k}^{\pm }\\ {\varvec{w}}_{k}^{\pm } \end{array}\right] \right) ={\mathbb {C}}^{\pm }\left[ \begin{array}{c} {\varvec{u}}_{k}^{\pm }\\ {\varvec{v}}_{k}^{\pm }\\ {\varvec{w}}_{k}^{\pm } \end{array}\right] \,, \end{aligned}$$

(113)

where

$$\begin{aligned} {\mathbb {C}}^{\pm }=\left[ \begin{array}{ccc} 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad {\mathbb {I}}\\ 0 &{}\quad {\mathbb {I}} &{}\quad 0 \end{array}\right] \left[ \begin{array}{ccc} {\mathbb {D}}^{\pm } &{}\quad 0 &{}\quad 0\\ 0 &{}\quad {\mathbb {D}}^{\pm } &{}\quad 0\\ 0 &{}\quad 0 &{}\quad {\mathbb {D}}^{\pm } \end{array}\right] +\left[ \begin{array}{ccc} 0 &{}\quad 0 &{}\quad {\mathbb {I}}\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 \end{array}\right] =\left[ \begin{array}{ccc} 0 &{}\quad 0 &{}\quad {\mathbb {I}}\\ 0 &{}\quad 0 &{}\quad {\mathbb {D}}^{\pm }\\ 0 &{}\quad {\mathbb {D}}^{\pm } &{}\quad 0 \end{array}\right] \,. \end{aligned}$$

(114)

We can rewrite the discretized PDE as

$$\begin{aligned} \left[ \begin{array}{c} {\varvec{u}}_{k+1}^{\pm }\\ {\varvec{v}}_{k+1}^{\pm }\\ {\varvec{w}}_{k+1}^{\pm } \end{array}\right] =\left( \Delta t{\mathbb {C}}^{\pm }+{\mathbb {I}}\right) \left[ \begin{array}{c} {\varvec{u}}_{k}^{\pm }\\ {\varvec{v}}_{k}^{\pm }\\ {\varvec{w}}_{k}^{\pm } \end{array}\right] =\left[ \begin{array}{c} {\varvec{s}}_{k}^{\pm }\\ {\varvec{y}}_{k}^{\pm }\\ {\varvec{z}}_{k}^{\pm } \end{array}\right] \,. \end{aligned}$$

(115)

To impose the BCs and JCs, we modify the equations as follows:

$$\begin{aligned} \left[ \begin{array}{cccc} 1 &{}\quad \cdots &{}\quad 0 &{} 0\\ \vdots &{}\quad \ddots &{}\quad \vdots &{} 0\\ 0 &{}\quad \cdots &{}\quad 1 &{} 0\\ &{} {\mathbb {D}}^{-}(N+1,:) \end{array}\right] \left[ \begin{array}{c} \\ {\varvec{u}}_{k+1}^{-}\\ \\ \end{array}\right]&= \left[ \begin{array}{c} 0\\ {\varvec{s}}_{k}^{-}(2:N)\\ {\mathbb {D}}^{+}(1,:){\varvec{u}}_{k}^{+} \end{array}\right] \,, \end{aligned}$$

(116)

$$\begin{aligned} \left[ \begin{array}{c} \\ {\varvec{u}}_{k+1}^{+}\\ \\ \end{array}\right]&= \left[ \begin{array}{c} u_{N-1,k+1}^{-}-g_{k}\\ {\varvec{s}}_{k}^{+}(2:N)\\ 0 \end{array}\right] \,, \end{aligned}$$

(117)

$$\begin{aligned} \left[ \begin{array}{c} {\varvec{v}}_{k+1}^{-}\\ \hline {\varvec{v}}_{k+1}^{+} \end{array}\right]&= \left[ \begin{array}{c} {\mathbb {D}}^{-}{\varvec{u}}_{k+1}^{-}\\ \hline {\mathbb {D}}^{+}{\varvec{u}}_{k+1}^{+} \end{array}\right] \,, \end{aligned}$$

(118)

$$\begin{aligned} \left[ \begin{array}{c} {\varvec{w}}_{k+1}^{-}\\ \hline {\varvec{w}}_{k+1}^{+} \end{array}\right]&= \left[ \begin{array}{c} 0\\ {\varvec{z}}_{k}^{-}(2:N+1)\\ \hline w_{N,k+1}^{-}-{\dot{g}}_{k+1}\\ {\varvec{z}}_{k}^{+}(2:N)\\ 0 \end{array}\right] . \end{aligned}$$

(119)

1.4 Elliptic PDEs

In this case we have no time evolution, and we simply need to solve $(({\mathbb {D}}^{\pm })^{2}+\mathrm{diag}(1/X_{i}^{\pm }){\mathbb {D}}^{\pm }){\varvec{u}}^{\pm }={\mathbb {M}}^{\pm }{\varvec{u}}^{\pm }=\mathbf{0}$, modified appropriately to account for the BCs and JCs. In particular, we first solve for ${\varvec{u}}^{+}$ using the BCs, and then for ${\varvec{u}}^{-}$ using the solution for ${\varvec{u}}^{+}$ to implement the JCs:

$$\begin{aligned} \left[ \begin{array}{c} {\mathbb {M}}^{+}(1:N-1,:)\\ 00\cdots 01\\ {\mathbb {D}}^{+}(N+1,:) \end{array}\right] {\varvec{u}}^{+}&= \left[ \begin{array}{c} \mathbf{0}(1:N-1)\\ 1-\tfrac{1}{2}\log (2L)\\ -\tfrac{1}{2L} \end{array}\right] \,, \end{aligned}$$

(120)

$$\begin{aligned} \left[ \begin{array}{c} {\mathbb {M}}^{-}(1:N-1,:)\\ 00\cdots 01\\ {\mathbb {D}}^{-}(N+1,:) \end{array}\right] {\varvec{u}}^{-}&= \left[ \begin{array}{c} \mathbf{0}(1:N-1)\\ u_{0}^{+}\\ {\mathbb {D}}^{+}(1,:){\varvec{u}}^{+}+1 \end{array}\right] \,. \end{aligned}$$

(121)

Exact Solution for the Elastic Wave Equation

A useful method for obtaining exact solutions to the problem (79) on ${\mathscr {I}}={\mathbb {R}}$ is outlined in Ref. [32]; we follow the same procedure here, except using a sinusoidal source time function (rather than a polynomial, as is done in Ref. [32]).

We begin by Fourier transforming the PDE in the spatial domain, using

$$\begin{aligned} {\hat{u}}\left( \xi ,t\right) =\int _{{\mathbb {R}}}\mathrm{d}x\,u\left( x,t\right) \mathrm{e}^{-\mathrm{i}\xi x}\,,\quad u\left( x,t\right) =\frac{1}{2\pi }\int _{{\mathbb {R}}}\mathrm{d}\xi \,{\hat{u}}\left( \xi ,t\right) \mathrm{e}^{\mathrm{i}\xi x}\,. \end{aligned}$$

(122)

Thus, multiplying the PDE in (79) by $\mathrm{e}^{-\mathrm{i}\xi x}$, integrating over ${\mathbb {R}}$ and applying integration by parts with the assumption of vanishing boundary terms, we get the following equation for the Fourier transform of u:

$$\begin{aligned} \ddot{{\hat{u}}}+\xi ^{2}{\hat{u}}=\mathrm{i}\xi g\left( t\right) \mathrm{e}^{-\mathrm{i}\xi x_{*}}\,. \end{aligned}$$

(123)

One can easily check that the exact solution of (123), with initial conditions ${\hat{u}}(\xi ,0)=0=\dot{{\hat{u}}}(\xi ,0)$, is simply

$$\begin{aligned} {\hat{u}}\left( \xi ,t\right) =\frac{1}{2}\mathrm{e}^{-\mathrm{i}\xi x_{*}}\sum _{\sigma =\pm }\sigma \mathrm{e}^{\sigma \mathrm{i}\xi t}\int _{0}^{t}\mathrm{d}\tau \,g\left( \tau \right) \mathrm{e}^{-\sigma \mathrm{i}\xi \tau }\,. \end{aligned}$$

(124)

Inserting $g(\tau )=\kappa \sin (\omega \tau )$ into (124) and carrying out the integrals, we get

$$\begin{aligned} {\hat{u}}=\frac{\mathrm{i}\kappa \mathrm{e}^{-\mathrm{i}\xi x_{*}}}{\xi ^{2}-\omega ^{2}}\left[ \xi \sin \left( \omega t\right) -\omega \sin \left( \xi t\right) \right] \,. \end{aligned}$$

(125)

Finally, plugging (125) back into (122) and using

$$\begin{aligned} \int _{{\mathbb {R}}}\mathrm{d}\xi \,\frac{\xi \mathrm{e}^{\mathrm{i}\xi \left( x-x_{*}\right) }}{\xi ^{2}-\omega ^{2}}&= \mathrm{i}\pi {\mathrm{sgn}}\left( x-x_{*}\right) \cos \left( \omega \left( x-x_{*}\right) \right) \,, \end{aligned}$$

(126)

$$\begin{aligned} \int _{{\mathbb {R}}}\mathrm{d}\xi \,\frac{\sin \left( \xi t\right) \mathrm{e}^{\mathrm{i}\xi \left( x-x_{*}\right) }}{\xi ^{2}-\omega ^{2}}&= \frac{\mathrm{i}\pi }{2\omega }\sum _{\sigma =\pm }\sigma {\mathrm{sgn}}\left( x-x_{*}+\sigma t\right) \sin \left( \omega \left( x-x_{*}+\sigma t\right) \right) \,, \end{aligned}$$

(127)

we get the solution (80).