The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

Abstract

The method of difference potentials was originally proposed by Ryaben’kii and can be interpreted as a generalized discrete version of the method of Calderon’s operators in the theory of partial differential equations. It has a number of important advantages; it easily handles curvilinear boundaries, variable coefficients, and non-standard boundary conditions while keeping the complexity at the level of a finite-difference scheme on a regular structured grid. The method of difference potentials assembles the overall solution of the original boundary value problem by repeatedly solving an auxiliary problem. This auxiliary problem allows a considerable degree of flexibility in its formulation and can be chosen so that it is very efficient to solve.

Compact finite difference schemes enable high order accuracy on small stencils at virtually no extra cost. The scheme attains consistency only on the solutions of the differential equation rather than on a wider class of sufficiently smooth functions. Unlike standard high order schemes, compact approximations require no additional boundary conditions beyond those needed for the differential equation itself. However, they exploit two stencils—one applies to the left-hand side of the equation and the other applies to the right-hand side of the equation.

We shall show how to properly define and compute the difference potentials and boundary projections for compact schemes. The combination of the method of difference potentials and compact schemes yields an inexpensive numerical procedure that offers high order accuracy for non-conforming smooth curvilinear boundaries on regular grids. We demonstrate the capabilities of the resulting method by solving the inhomogeneous Helmholtz equation with a variable wavenumber with high order (4 and 6) accuracy on Cartesian grids for non-conforming boundaries such as circles and ellipses.

Appendix A: Coordinates Associated with a Curve and Equation-Based Extension

1.1 A.1 Elliptical Coordinates

Let Γ be an ellipse with semi-axes a and b:

$$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1. $$

(69)

Then, the equation-based extension (introduced in Sect. 4.2 for the case of a circle) is convenient to build using the elliptical coordinates.

Denote $d=\sqrt{a^{2}-b^{2}}$ the distance from the center of the ellipse (69) to either of its foci. A common definition of elliptical coordinates (η,φ) is given by:

$$ \left \{ \begin{array}{l} x=d\cosh\eta\cos\varphi,\\[3pt] y=d\sinh\eta\sin\varphi, \end{array} \right . $$

(70)

where η≥0 and φ∈[0,2π). The coordinate lines that correspond to (70) are families of ellipses and hyperbolas on the plane, see Fig. 7. For a fixed η=η ₀, the coordinate line is an ellipse:

$$ \frac{x^2}{d^2\cosh^2\eta_0}+\frac{y^2}{d^2\sinh^2\eta_0}=1, $$

so that the original ellipse (69) corresponds to

$$ \eta_0=\frac{1}{2}\ln \frac{a+b}{a-b}. $$

(71)

For a fixed φ=φ ₀, the coordinate line is a hyperbola:

$$ \frac{x^2}{d^2\cos^2\varphi_0}-\frac{y^2}{d^2\sin^2\varphi_0}=\cosh^2\eta- \sinh^2\eta=1. $$

The basis vectors for elliptical coordinates are (see Fig. 7):

$$ \boldsymbol {e}_1\equiv\hat{\eta}=\frac{\partial \boldsymbol {r}}{\partial\eta}=(d\sinh\eta\cos \varphi,d \cosh\eta\sin\varphi) $$

and

$$ \boldsymbol {e}_2\equiv\hat{\varphi}=\frac{\partial \boldsymbol {r}}{\partial\varphi}=(-d\cosh\eta \sin\varphi,d \sinh\eta\cos\varphi), $$

where r=(x,y) is the radius-vector defined via (70). Hence, the Lame coefficients are equal to one another:

$$ H_1\equiv H_\eta=| \boldsymbol {e}_1|=d\sqrt{\sinh^2\eta+\sin^2\varphi}=| \boldsymbol {e}_2|=H_2\equiv H_\varphi. $$

(72)

Accordingly, the Helmholtz equation (9) in the elliptical coordinates (70) transforms into

$$ \frac{1}{H^2} \biggl[ \frac{\partial^2 u}{\partial\eta^2} +\frac{\partial^2 u}{\partial\varphi^2} \biggr]+k^2(\eta,\varphi)u=f, $$

(73)

where H=H _η =H _φ , see formula (72).

Fig. 7

Elliptical coordinates

Suppose that $\pmb {\xi }_{\varGamma}=(\xi_{0},\xi_{1}) |_{\varGamma}$ is given on the ellipse (69) so that ξ ₀=ξ ₀(φ) and ξ ₁=ξ ₁(φ). In the vicinity of this ellipse, we define a new smooth function v=v(η,φ) by means of the Taylor formula [cf. formula (42)]:

$$ v(\eta,\varphi)=v(\eta_0, \varphi)+\sum_{l=1}^L\frac{1}{l!} \frac{\partial^l v(\eta_0,\varphi)}{\partial\eta^l}(\eta-\eta_0)^l, $$

(74)

where η ₀ is given by expression (71), and the choice of L is discussed in Sect. 4.3. The zeroth and first order derivatives in formula (74) are obtained by requiring that [cf. formula (43)]

$$ v(\eta_0,\varphi)= \xi_0(\varphi)\quad\text{and}\quad\frac{\partial v(\eta_0,\varphi)}{\partial\eta}= \xi_1(\varphi). $$

(75)

Note that whereas in polar coordinates $\frac{\partial v}{\partial n}=\frac{\partial v}{\partial r}$, in elliptical coordinates we have: $\frac{\partial v}{\partial n}=\frac{1}{H}\frac{\partial v}{\partial\eta}$. The higher order derivatives for formula (74) are obtained via equation-based differentiation:

and

where

The function v(η,φ) of (74) evaluated at the nodes of the grid boundary γ is called the equation-based extension of $\pmb {\xi }_{\varGamma}$ from the ellipse Γ given by (69) to γ [cf. formula (48)]:

$$ \xi_\gamma=\operatorname {\mbox {\textbf {\textit {Ex}}}}\pmb {\xi }_\varGamma\stackrel{\text{def}}{=}v(\eta,\varphi) |_\gamma. $$

1.2 A.2 General Case

Equation-based extensions can also be built without assuming any special regular shape of Γ, such as a circle or an ellipse. Suppose that Γ is a general non-self-intersecting smooth closed curve on the plane, and that it is parameterized by its arc length s:

$$ \varGamma=\bigl\{\boldsymbol {R}(s)|0\le s\le S\bigr\}, $$

where R is the radius-vector that traces the curve. Assume for definiteness that as s increases the point R(s) moves counterclockwise along Γ. The unit tangent vector to Γ is given by

$$ \pmb {\tau }=\pmb {\tau }(s)=\frac{d\boldsymbol {R}}{d s}. $$

(76)

In addition to the unit tangent, we also consider the unit normal to the curve Γ:

$$ \pmb {\nu }=(\nu_x,\nu_y)=( \tau_y,-\tau_x). $$

(77)

Given a counterclockwise parametrization R=R(s), the normal $\pmb {\nu }$ of (77) will always be pointing outward with respect to the domain Ω, and hence the pair of vectors $(\pmb {\nu },\pmb {\tau })$ will always have a fixed right-handed orientation on the plane.

Note, that there is a simple relation between the tangent (76), the normal (77), and the curvature κ of the curve Γ. It is given by the Frenet formula:

$$ \frac{d\pmb {\tau }}{ds}=\kappa \pmb {\nu }. $$

(78)

The vector $\frac{d\pmb {\tau }}{ds}$ is directed toward the center of curvature, i.e., it may point either toward Ω or away from Ω depending on which direction the curve bends. Since the normal (77) has a fixed orientation, the curvature κ=κ(s) in formula (78) should be taken with the sign (see, e.g., [32, Part 1]):

$$ \kappa(s)= \begin{cases} \vert \frac{d\pmb {\tau }}{ds}\vert , & \text{if}\ \frac{d\pmb {\tau }}{ds}\cdot \pmb {\nu }>0,\\[3pt] -\lvert \frac{d\pmb {\tau }}{ds}\rvert , & \text{if}\ \frac{d\pmb {\tau }}{ds}\cdot \pmb {\nu }<0. \end{cases} $$

(79)

We can now define the coordinates associated with the curve Γ. First, we re-emphasize that since Γ is smooth, both the tangent $\pmb {\tau }=\pmb {\tau }(s)$ and the normal $\pmb {\nu }=\pmb {\nu }(s)$ are smooth functions of s. Consider a point on the plane, which is a given node from the grid boundary γ. Draw the shortest normal from this point onto the curve Γ. Suppose that the value of the parameter of the curve at the foot of this normal is s, and the distance between the original point and the foot of the normal is n, see Fig. 8. As the position of the point may be on either side of the curve, the value of the distance n is taken with the sign: n>0 corresponds to the positive direction $\pmb {\nu }$, i.e., to the exterior of Ω, and n<0 corresponds to the negative direction of $\pmb {\nu }$, i.e., to the interior of Ω. The pair of numbers (n,s) provides the coordinates that identify the location of a given point on the plane. These coordinates are obviously orthogonal.

Fig. 8

New coordinates (n,s) and equation-based extension

A limitation of the coordinates (n,s) is that for a general shape of the boundary Γ there may be some ambiguity, as it is possible that multiple shortest normals will exist for a given node. Hence, the distance function will be multi-valued and not differentiable w.r.t. the arc length s. This may happen, in particular, when the boundary Γ has a “cavity”. An obvious sufficient condition that avoids such ambiguities is to require that the coordinates (n,s) be used only inside a curvilinear strip of width $2\bar{R}$ that straddles Γ, where $\bar{R}=\min_{s} R(s)$ is the minimum radius of curvature. In other words, we require that if $\frac{d\pmb {\tau }}{ds}\cdot \pmb {\nu }>0$ then $n<\bar{\kappa}^{-1}\equiv\bar{R}$, and if $\frac{d\pmb {\tau }}{ds}\cdot \pmb {\nu }<0$ then $n>\bar{\kappa}^{-1}\equiv-\bar{R}$. This limitation, however, is not severe, because we use the coordinates (n,s) only for the points of the grid boundary γ, which are all about one grid size h away from the curve Γ, see Figs. 3 and 8. Having multiple shortest distances for a given node would then imply that the minimum radius of curvature $\bar{R}$ is also of order h. This means, in turn, that the grid does not adequately resolve the geometry, and needs to be refined. Let us also note that the simulations in this paper, see Sect. 5, do not involve any shapes with potential cavities, i.e., non-convex features with the curvature κ∼h ⁻¹. In the future, we will analyze shapes with “small” features.

The coordinates (n,s) are orthogonal but not orthonormal. For a given point (n,s), its radius-vector r is expressed as follows:

and consequently, the basis vectors are given by

and

where we have used formulae (76), (77), (78), and (79). Accordingly, the Lame coefficients for the coordinates (n,s) are

and

(80)

where the last equality in (80) holds because n<κ ⁻¹ for κ>0 and n>κ ⁻¹ for κ<0.

In the coordinates (n,s), (9) becomes

$$ \frac{1}{H_s} \biggl[\frac{\partial}{\partial n} \biggl(H_s\frac{\partial u}{\partial n} \biggr) +\frac{\partial}{\partial s} \biggl( \frac{1}{H_s}\frac{\partial u}{\partial s} \biggr) \biggr]+k^2(n,s)u=f, $$

(81)

where H _s =H _s (n,s) is given by (80), and where we have taken into account that H _n ≡1. Equation (81) will be used for building the equation-based extension of a given $\pmb {\xi }_{\varGamma}$ from the continuous boundary Γ to the nodes of the grid boundary γ similar to how the polar Helmholtz equation (41) was used in Sect. 4.2 for building the corresponding extension from the circle.

Note, that if Γ is a circle of radius R, then the foregoing general constructs transform into the corresponding constructs for polar coordinates described in Sect. 4.2. Indeed, in this case the curvature κ of (79) does not depend on s:

$$ \kappa=-\frac{1}{R}, $$

and consequently [see formula (80)],

$$ H_s=1+\frac{n}{R}=\frac{R+n}{R}=\frac{r}{R}. $$

Then, according to (81) we can write:

$$ \Delta u= \frac{R}{r} \biggl[\frac{\partial}{\partial n} \biggl( \frac {r}{R}\frac{\partial u}{\partial n} \biggr) +\frac{\partial}{\partial s} \biggl( \frac{R}{r}\frac{\partial u}{\partial s} \biggr) \biggr] =\frac{1}{r} \frac{\partial}{\partial n} \biggl(r\frac{\partial u}{\partial n} \biggr) +\frac{R^2}{r^2} \frac{\partial^2 u}{\partial s^2}. $$

Finally, we have n=r−R so that $\frac{\partial}{\partial n}=\frac {\partial}{\partial r}$, and s=Rθ so that $\frac{\partial}{\partial s}=\frac{1}{R}\frac {\partial}{\partial\theta}$, which yields:

$$ \Delta u=\frac{1}{r}\frac{\partial}{\partial r} \biggl(r\frac{\partial u}{\partial r} \biggr) +\frac{1}{r^2}\frac{\partial^2 u}{\partial\theta^2}. $$

Let $\pmb {\xi }_{\varGamma}=(\xi_{0}(s),\xi_{1}(s))$ be given on Γ. In the vicinity of Γ, we define a new smooth function v=v(n,s) by means of the Taylor formula [cf. formula (42)]:

$$ v(n,s)=v(0,s)+\sum_{l=1}^L \frac{1}{l!}\frac{\partial^l v(0,s)}{\partial n^l}n^l. $$

(82)

The zeroth and first order derivatives in formula (82) are obtained by requiring that $\operatorname {\mbox {\textbf {\textit {Tr}}}}v=\pmb {\xi }_{\varGamma}$:

$$ v(0,s)=\xi_0(s)\quad\text{and}\quad \frac{\partial v(0,s)}{\partial n}=\xi_1(s). $$

(83)

All higher order derivatives in formula (82) are determined with the help of equation (81) applied to v. For convenience, we multiply both sides by H _s and then solve for the second derivative with respect to n, which yields:

$$ \frac{\partial^2 v}{\partial n^2} =\tilde{f}-\tilde{k}^2(n,s)v-\frac{\partial}{\partial s} \biggl(\frac{1}{H_s} \frac{\partial v}{\partial s} \biggr)-\frac{\partial H_s}{\partial n}\frac{\partial v}{\partial n}, $$

(84)

where $\tilde{k}=H_{s}k$, $\tilde{f}=H_{s}f$, and $\frac{\partial H_{s}}{\partial n}=-\kappa(s)$, see formula (80). Consequently,

$$ \frac{\partial^2 v(0,s)}{\partial n^2} =\tilde{f}(0,s)-\tilde{k}^2(0,s)\xi_0(s)-\frac{\partial}{\partial s} \biggl( \frac{1}{H_s}\frac{\partial\xi_0(s)}{\partial s} \biggr)+\kappa (s)\xi_1(s). $$

(85)

Next, we differentiate equation (84) with respect to n:

$$ \frac{\partial^3 v}{\partial n^3} =\frac{\partial\tilde{f}}{\partial n}- \frac{\partial\tilde{k}^2}{\partial n}v -\tilde{k}^2\frac{\partial v}{\partial n} - \frac{\partial}{\partial s} \biggl(\frac{\partial H_s^{-1}}{\partial n}\frac{\partial v}{\partial s} \biggr) - \frac{\partial}{\partial s} \biggl(\frac{1}{H_s}\frac{\partial}{\partial s} \biggl[ \frac{\partial v}{\partial n} \biggr] \biggr) +\kappa\frac{\partial^2 v}{\partial n^2}, $$

(86)

and substituting v(0,s) and $\frac{\partial v(0,s)}{\partial n}$ from (83) and $\frac{\partial^{2} v(0,s)}{\partial n^{2}}$ from (85), we obtain $\frac{\partial^{3} v(0,s)}{\partial n^{3}}$. Similarly, the fourth normal derivative $\frac{\partial^{4} v(0,s)}{\partial n^{4}}$ can be evaluated by differentiating equation (86) with respect to n:

and then substituting the previously computed derivatives v(0,s), $\frac{\partial v(0,s)}{\partial n}$, $\frac{\partial^{2} v(0,s)}{\partial n^{2}}$, and $\frac{\partial^{3} v(0,s)}{\partial n^{3}}$. Higher order derivatives (e.g., for the sixth order scheme) can be obtained in the same manner.

As in (42), we emphasize that formula (82) is not an approximation of a given v(n,s) by its truncated Taylor’s expansion. Rather it is the definition of a new function v(n,s). This function is used for building the equation-based extension of $\pmb {\xi }_{\varGamma}$ from Γ to γ:

$$ \xi_\gamma=\operatorname {\mbox {\textbf {\textit {Ex}}}}\pmb {\xi }_\varGamma\stackrel{ \text{def}}{=}v(n,s) |_\gamma. $$

(87)

So, extension (87) is obtained by drawing a normal from a given node of γ to Γ, see Fig. 8, and then using the Taylor formula with higher order derivatives computed by differentiating the governing equation (81). If Γ is a circle, then the derivatives with respect to n in formula (82) become radial derivatives, and accordingly, the extension (87) transforms into (48).

Extension of the right-hand side f is built with the help of the Taylor formula

$$ f(n,s)=f(0,s)+\sum_{l=1}^{L-2} \frac{1}{l!}\frac{\partial^l f(0,s)}{\partial n^l}n^l. $$

(88)

The derivatives of f that enter into (88) can be computed as one-sided normal derivatives on the interior side of Γ, similarly to how it is done for formula (49) when Γ is a circle.

Note that for more complicated shapes it may often be convenient to have a piece-wise parametrization for the curve Γ as opposed to the global parametrization. Piece-wise parametrization and different bases (40a) for different portions of the boundary (e.g., independent Chebyshev systems) may also pave the way toward discontinuous coefficients in the boundary conditions, non-smooth boundaries, and other formulations which will be analyzed in the future, see Sect. 6.

It should finally be mentioned that a somewhat more elaborate procedure will yield an equation-based extension for an equation more general than (9):

$$ \mathrm {div}(\varepsilon\cdot \mathrm {grad}\,u)+k^2u=f. $$

(89)

In formula (89), ε=ε(x,y) is a symmetric and positive definite tensor of rank 2. Equation (9) is a particular case of (89) where ε is the identity. In [6], we have considered another particular case of (89), with ε being diagonal in the Cartesian coordinates: ε=diag{a,b}.