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A Fourth-Order Kernel-Free Boundary Integral Method for the Modified Helmholtz Equation

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Abstract

Based on the kernel-free boundary integral method proposed by Ying and Henriquez (J Comput Phys 227(2):1046–1074, 2007), which is a second-order accurate method for general elliptic partial differential equations, this work develops it to be a fourth-order accurate version for the modified Helmholtz equation. The updated method is in line with the original one. Unlike the traditional boundary integral method, it does not need to know any analytical expression of the fundamental solution or Green’s function in evaluation of boundary or volume integrals. Boundary value problems under consideration are reformulated into Fredholm boundary integral equations of the second kind, whose corresponding discrete forms are solved with the simplest Krylov subspace iterative method, the Richardson iteration. During each iteration, a Cartesian grid based nine-point compact difference scheme is used to discretize the simple interface problem whose solution is the boundary or volume integral in the BIEs. The resulting linear system is solved by a fast Fourier transform based solver, whose computational work is roughly proportional to the number of grid nodes in the Cartesian grid used. As the discrete boundary integral equations are well-conditioned, the iteration converges within an essentially fixed number of steps, independent of the mesh parameter. Numerical results are presented to verify the solution accuracy and demonstrate the algorithm efficiency.

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Authors and Affiliations

  1. School of Mathematical Sciences, Shanghai Jiao Tong University, Minhang, Shanghai, 200240, People’s Republic of China

    Yaning Xie

  2. School of Mathematical Sciences, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong University, Minhang, Shanghai, 200240, People’s Republic of China

    Wenjun Ying

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  1. Yaning Xie
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  2. Wenjun Ying
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Correspondence to Wenjun Ying.

Additional information

The original version of this article was revised: The errors in the equations, texts and references have been corrected.

Research of the second author was supported in part by the National Science Foundation of the USA under Grant DMS 0915023, and is supported by the National Natural Science Foundation of China under Grants DMS 91330203, DMS 11101278, DMS 91130012 and DMS 11771290. Research of the second author was also supported by the Young Thousand Talents Program of China.

Appendix: Calculation for Jumps of Partial Derivatives

Appendix: Calculation for Jumps of Partial Derivatives

Jumps of partial derivatives of the solution to the interface problem (22) are needed not just in the correction for the finite difference equations but also in the extraction of boundary data from the discrete finite difference solutions.

Let \(u(\mathbf{x})\) be a piecewise smooth function defined in \({\mathcal B}\), whose normal derivative \(u_{\mathbf{n}}\) as well as u itself may be discontinuous across the interface. Computation of jumps for u on \(\varGamma \) starts from the interface problem (22), which is reiterated below

$$\begin{aligned} {\triangle }u -\kappa u= & {} \tilde{f} \quad \text {in }\,\, {\mathcal B}\setminus \varGamma , \end{aligned}$$
(40)
$$\begin{aligned}{}[u]= & {} \varphi \quad \text {on }\,\, \varGamma , \end{aligned}$$
(41)
$$\begin{aligned}{}[u_{\mathbf{n}}]= & {} \psi \quad \text {on }\,\, \varGamma . \end{aligned}$$
(42)

Let s be the arc length parameter of the curve \(\varGamma \), \(\mathbf{t} = (x'(s), y'(s))^\mathrm{T}\) be the unit tangent vector, \(\mathbf{n} = (y'(s), -x'(s))^\mathrm{T} = \mathbf{t}^{\perp }\) be the unit outward normal vector on \(\varGamma \). Suppose that \(\varphi = \varphi (s)\) and \(\psi = \psi (s)\) are two sufficiently smooth functions defined on \(\varGamma \).

Differentiating (41) with respect to the arc length parameter s, together with (42), we get

$$\begin{aligned} \left\{ \begin{array}{l} x'(s) [u_x] + y'(s) [u_y] = \varphi _s \\ y'(s) [u_x] - x'(s) [u_y] = \psi \end{array} \right. . \end{aligned}$$
(43)

Solving this two by two linear system gives us jumps of the first-order partial derivatives of u on \(\varGamma \).

Differentiating the identities in (43) with respect to the arc length parameter s, respectively, together with the modified Helmholtz equation (40), we get

$$\begin{aligned} \left\{ \begin{array}{l} (x')^2 [u_{xx}] + 2 x' y' [u_{xy}] + (y')^2 [u_{yy}] = \varphi _{ss} - \big \{ x''(s) [u_x] + y''(s) [u_y] \big \} \\ x' y' [u_{xx}] + \big \{ (y')^2 - (x')^2 \big \} [u_{xy}] - x'y' [u_{yy}] = \psi _s - y'' [u_x] + x'' [u_y] \\ {[} u_{xx} ] + [ u_{yy} ] = f +\kappa [u] \end{array} \right. . \end{aligned}$$
(44)

Solving this three by three linear system gives us jumps of the second-order partial derivatives of u on \(\varGamma \).

Differentiating the first equation in (44) with respect to the arc length parameter s gives us

$$\begin{aligned} (x')^3 [u_{xxx}] + 3 (x')^2 y' [u_{xxy}] + 3 x' (y')^2 [u_{xyy}] + (y')^3 [u_{yyy}] = r_{3,1} \end{aligned}$$
(45)

with

$$\begin{aligned} r_{3,1} = \varphi _{sss} - (x''' [u_x] + y''' [u_y]) - 3 \bigl \{ x''x' [u_{xx}] + (x'' y' + x' y'') [u_{xy}] + y'' y' [u_{yy}] \bigr \}. \end{aligned}$$

Differentiating the second equation in (44) with respect to the arc length parameter s gives us

$$\begin{aligned}&(x')^2 y' [u_{xxx}] + \bigl \{ 2 x' (y')^2 - (x')^3\bigr \} [u_{xxy}] \nonumber \\&\quad + \bigl \{ (y')^3 - 2 (x')^2 y' \bigr \} [u_{xyy}] - x' (y')^2 [u_{yyy}] = r_{3,2} \nonumber \\ \end{aligned}$$
(46)

with

$$\begin{aligned} r_{3,2}= & {} \psi _{ss} - y''' [u_x] + x''' [u_y] - (x''y' + 2 x' y'' ) [u_{xx}] \\&+ 3 ( x' x'' - y'y'') [u_{xy}] + (2 x'' y' + x' y'') [u_{yy}]. \end{aligned}$$

We may get two more equations for jumps of the third-order partial derivatives by first differentiating the modified Helmholtz equation and then taking jumps of the resulting equations across \(\varGamma \). They read

$$\begin{aligned} {[}u_{xxx}] + [u_{xyy}]= & {} f_x+\kappa \, [u_x], \end{aligned}$$
(47)
$$\begin{aligned} {[}u_{xxy}] + [u_{yyy}]= & {} f_y+\kappa \, [u_y]. \end{aligned}$$
(48)

We may get jumps of the third-order partial derivatives, \([u_{xxx}]\), \([u_{xxy}]\), \([u_{xyy}]\) and \([u_{yyy}]\), by solving the system consisting of Eqs. (45)–(48).

Differentiating (45) with respect to the arc length parameter s yields

$$\begin{aligned}&(x')^4 [u_{xxxx}] + 4 (x')^3 y' [u_{xxxy}] + 6 (x')^2 (y')^2 [u_{xxyy}] \nonumber \\&\quad +\,4 x' (y')^3 [u_{xyyy} + (y')^4 [u_{yyyy}] = r_{4,1} \nonumber \\ \end{aligned}$$
(49)

with

$$\begin{aligned} r_{4,1}= & {} \varphi _{ssss} - (x''''[u_x] + y''''[u_y]) -\bigl \{ (4 x'''x' + 3 x'' x'') [u_{xx}] \nonumber \\&+ \, (4 x''' y' + 6 x'' y'' + 4 x' y''') [u_{xy}] + (4 y''' y' + 3 (y'')^2) [u_{yy}] \bigr \}\nonumber \\&- \, 6 \bigl \{x''(x')^2 [u_{xxx}]+(2 x' x'' y' + (x')^2 y'') [u_{xxy}] \nonumber \\&+\, (x'' (y')^2 + 2 x' y' y'') [u_{xyy}]+ y'' (y')^2 [u_{yyy}] \bigr \} . \end{aligned}$$

Differentiating (46) with respect to the arc length parameter s leads to

$$\begin{aligned}&(x')^3 y' [u_{xxxx}]+(x')^2 \bigl \{3 (y')^2 - (x')^2 \bigr \} [u_{xxxy}]+ 3 x' y' \bigl \{(y')^2 - (x')^2 \bigr \} [u_{xxyy}] \nonumber \\&\quad + \, (y')^2 \bigl \{(y')^2 - 3 (x')^2 \bigr \} [u_{xyyy}] - x' (y')^3 [u_{yyyy}] = r_{4,2} \nonumber \\ \end{aligned}$$
(50)

with

$$\begin{aligned} r_{4,2}= & {} \psi _{sss} - y'''' [u_x] + x'''' [u_y] - (x'''y' + 3x''y'' + 3 x'y''' ) [u_{xx}] \\&+ \, ( 3 x'' x'' + 4 x' x''' - 3 y''y'' - 4 y' y''') [u_{xy}] + (3 x''' y' + 3 x'' y'' + x' y''') [u_{yy}] \\&- 3 \bigl \{ x' x'' y' + (x')^2 y'' \bigr \} [u_{xxx}] + 3 \bigl \{ 2 (x')^2 x'' - x'' (y')^2 - 3 x' y' y'' \bigr \} [u_{xxy}] \\&- 3 \bigl \{ 2 (y')^2 y'' - 3 x' x'' y' - (x')^2 y'' \bigr \} [u_{xyy}] + 3 \bigl \{ x'' (y')^2 + x' y' y'' \bigr \} [u_{yyy}]. \end{aligned}$$

Other three equations for jumps of the fourth-order partial derivatives read

$$\begin{aligned}{}[u_{xxxx}] + [u_{xxyy}]= & {} f_{xx}+\kappa [u_{xx}], \end{aligned}$$
(51)
$$\begin{aligned}{}[u_{xxxy}] + [u_{xyyy}]= & {} f_{xy}+\kappa [u_{xy}], \end{aligned}$$
(52)
$$\begin{aligned}{}[u_{xxyy}] + [u_{yyyy}]= & {} f_{yy}+\kappa [u_{yy}], \end{aligned}$$
(53)

which are similarly obtained by first differentiating the modified Helmholtz equation and then taking jumps of the resulting equations across \(\varGamma \).

We may get jumps of the fourth-order partial derivatives, \([u_{xxxx}]\), \([u_{xxxy}]\), \([u_{xxyy}]\), \([u_{xyyy}]\) and \([u_{yyyy}]\), by solving the system consisting of Eqs. (49)–(53).

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Xie, Y., Ying, W. A Fourth-Order Kernel-Free Boundary Integral Method for the Modified Helmholtz Equation. J Sci Comput 78, 1632–1658 (2019). https://doi.org/10.1007/s10915-018-0821-8

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