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Spatially Dispersionless, Unconditionally Stable FC–AD Solvers for Variable-Coefficient PDEs

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Abstract

We present fast, spatially dispersionless and unconditionally stable high-order solvers for partial differential equations (PDEs) with variable coefficients in general smooth domains. Our solvers, which are based on (i) A certain “Fourier continuation” (FC) method for the resolution of the Gibbs phenomenon on equi-spaced Cartesian grids, together with (ii) A new, preconditioned, FC-based solver for two-point boundary value problems for variable-coefficient Ordinary Differential Equations, and (iii) An Alternating Direction strategy, generalize significantly a class of FC-based solvers introduced recently for constant-coefficient PDEs. The present algorithms, which are applicable, with high-order accuracy, to variable-coefficient elliptic, parabolic and hyperbolic PDEs in general domains with smooth boundaries, are unconditionally stable, do not suffer from spatial numerical dispersion, and they run at Fast Fourier Transform speeds. The accuracy, efficiency and overall capabilities of our methods are demonstrated by means of applications to challenging problems of diffusion and wave propagation in heterogeneous media.

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References

  1. Albin, N., Bruno, O.P.: A spectral FC solver for the compressible Navier-Stokes equations in general domains I: explicit time-stepping. J. Comput. Phys. 230(16), 6248–6270 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, vol. 1. Springer, Berlin (1978)

    Google Scholar 

  3. Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover Publications, New York (2001)

    MATH  Google Scholar 

  4. Boyd, J.P.: A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds. J. Comput. Phys. 178(1), 118–160 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bruno, O.P., Han, Y., Pohlman, M.M.: Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis. J. Comput. Phys. 227(2), 1094–1125 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bruno, O.P., Lyon, M.: High-order unconditionally stable FC–AD solvers for general smooth domains I. Basic elements. J. Comput. Phys. 229(6), 2009–2033 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bruno, O.P.: Fast, high-order, high-frequency integral methods for computational acoustics and electromagnetics. In Ainsworth, M., Davies, P., Duncan, D., Martin, P., Rynne, B. (eds.) Topics in Computational Wave Propagation Direct and Inverse Problems Series, Volume 31 of Lecture Notes in Computational Science and, Engineering, pp. 43–82 (2003)

  8. Douglas Jr, J., Rachford Jr, H.H.: On the numerical solution of heat conduction problems in two and three space variables. T. Am. Math. Soc. 82(2), 421–439 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  9. Evans, L.C.: Partial Differential Equations, Volume 12 of Graduate Studies in Mathematics. American Mathematical Society (1998)

  10. Frigo, M., Johnson, S.G.: The design and implementation of FFTW 3. Proc. IEEE Micr. Elect. 93(2), 216–231 (2005)

    Google Scholar 

  11. Hesthaven, J., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  12. Huybrechs, D.: On the Fourier extension of non-periodic functions. SIAM J. Numer. Anal. 47(6), 4326–4355 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lyon, M., Bruno, O.P.: High-order unconditionally stable FC–AD solvers for general smooth domains II. Elliptic, parabolic and hyperbolic PDEs; theoretical considerations. J. Comput. Phys. 229(9), 3358–3381 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Marchuk, G.I.: Finite difference methods: splitting and alternating direction methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. 1, pp. 197–462. North-Holland, Amsterdam (1990)

    Google Scholar 

  15. Peaceman, D.W., Rachford Jr, H.H.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3(1), 28–41 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  16. Sun, W., Huang, W., Russell, R.D.: Finite difference preconditioning for solving orthogonal collocation equations for boundary value problems. SIAM J. Numer. Anal. 33(6), 2268–2285 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Walker, H.F.: Implementation of the GMRES method using Householder transformations. SIAM J. Sci. Stat. Comput. 9, 152–163 (1988)

    Article  MATH  Google Scholar 

  18. Weinberger, H.F.: A First Course in Partial Differential Equations with Complex Variables and Transform Methods. Dover, New York (1995)

    Google Scholar 

Download references

Acknowledgments

The authors gratefully acknowledge support from NSF and AFOSR. The work of A. Prieto was partially supported by Ministerio de Educación y Ciencia of Spain under project grant MTM2008-02483, programme Angeles Alvariño (grant 2007/AA-076) and programme Juan de la Cierva (grant JCI-2010-06793).

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Authors and Affiliations

  1. Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, 91125, USA

    O. P. Bruno

  2. Departamento de Matemáticas, Universidade da Coruña, 15071 , A Coruña, Spain

    A. Prieto

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  1. O. P. Bruno
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  2. A. Prieto
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Correspondence to O. P. Bruno.

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Appendix: An Auxiliary Lemma

Appendix: An Auxiliary Lemma

Lemma 1

Let \(\tilde{q}^{\ell }, g_{a}\) and \(g_{b}\) be smooth functions defined in the interval \([b,c]\), and let \(\tilde{q}^{\ell }\) be strictly positive in that interval. If \(g_{a}\) and \(g_{b}\) satisfy the conditions (25), then the overdetermined ODE problem

$$\begin{aligned}&v-\tilde{p}\frac{dv}{dx} -\tilde{q}^{\ell }\frac{d^{2}v}{dx^2}=g_{a}+\mu g_{b} \quad \mathrm{in}\ (b,c),\end{aligned}$$
(42)
$$\begin{aligned}&v(b)=\frac{dv}{dx}(b)=0, \quad v(c)=\frac{dv}{dx}(c)=0, \end{aligned}$$
(43)

is not solvable: Eqs. (42)–(43) do not admit solutions \(v\) for any real value of the constant \(\mu \).

Proof

Assume a solution \(v\) of the problem (42)–(43) exists. Denoting by \(G(x,\xi )\) the Green function of the problem,

$$\begin{aligned}&G(x,\xi )-\tilde{p}\frac{\partial }{\partial x} G(x,\xi ) -\tilde{q}^{\ell }\frac{\partial ^{2}}{\partial x^2}G(x,\xi )=\delta _{(x=\xi )} \quad \text{ in } (b,c),\\&G(b,\xi )=G(c,\xi )=0, \end{aligned}$$

and letting

$$\begin{aligned} h=g_{a}+\mu g_{b}, \end{aligned}$$
(44)

the solution \(v\) can be expressed in the form

$$\begin{aligned} v(x)=\int \limits _{b}^{c}G(x,\xi )h(\xi )\,\mathrm{d}\xi . \end{aligned}$$

Taking into account the Neumann boundary conditions (43) we then obtain

$$\begin{aligned} \frac{dv}{dx}(b)&= \int \limits _{b}^{c}\frac{\partial G}{\partial x}(b,\xi )h(\xi )\,\mathrm{d}\xi =0,\end{aligned}$$
(45)
$$\begin{aligned} \frac{dv}{dx}(c)&= \int \limits _{b}^{c}\frac{\partial G}{\partial x}(c,\xi )h(\xi )\,\mathrm{d}\xi =0. \end{aligned}$$
(46)

Now, as is known (see e.g. in [18, Ch. V.28]), the function \(\partial G/\partial \xi \) satisfies the ODE problems

$$\begin{aligned}&\frac{\partial G}{\partial \xi }(x,b)- \tilde{p}\frac{\partial }{\partial x}\left( \frac{\partial G}{\partial \xi }(x,b)\right) -\tilde{q}^{\ell } \frac{\partial ^{2}}{\partial x^2} \left( \frac{\partial G}{\partial \xi }(x,b)\right) =0 \quad \text{ in } (b,c),\\&\frac{\partial G}{\partial \xi }(b,b)=\frac{1}{\tilde{q}^{\ell }(b)},\quad \frac{\partial G}{\partial \xi }(c,b)=0 \end{aligned}$$

and

$$\begin{aligned}&\frac{\partial G}{\partial \xi }(x,c)- \tilde{p}\frac{\partial }{\partial x}\left( \frac{\partial G}{\partial \xi }(x,c)\right) -\tilde{q}^{\ell }\frac{\partial ^{2}}{\partial x^2} \left( \frac{\partial G}{\partial \xi }(x,c)\right) =0 \quad \text{ in } (b,c),\\&\frac{\partial G}{\partial \xi }(b,c)=0,\quad \frac{\partial G}{\partial \xi }(c,c)=-\frac{1}{\tilde{q}^{\ell }(c)}. \end{aligned}$$

In view of the identity \(\frac{\partial G}{\partial x}(x,\xi )=\frac{\partial G}{\partial \xi }(\xi ,x)\) (which follows from the symmetry \(G(x,\xi )=G(\xi ,x)\) of the Green function) it follows that the function

$$\begin{aligned} H (x) = \frac{\partial G}{\partial x}(b,x)- \frac{\partial G}{\partial x}(c,x) \end{aligned}$$

satisfies the two-point boundary-value problem

$$\begin{aligned}&H- \tilde{p}\frac{dH}{dx} -\tilde{q}^{\ell }\frac{d^{2}H}{d x^2}=0 \quad \text{ in } (b,c),\\&H(b)=\frac{1}{\tilde{q}^{\ell }(b)},\quad H(c)=\frac{1}{\tilde{q}^{\ell }(c)}. \end{aligned}$$

Applying the strong maximum principle [9] to this elliptic equation we obtain the estimate

$$\begin{aligned} H (x) = \frac{\partial G}{\partial x}(b,x)-\frac{\partial G}{\partial x}(c,x) \ge C > 0\quad \text{ for } x\in [b,c], \end{aligned}$$

where \(C\) is the strictly positive constant \(C=\min \{1/\tilde{q}^{\ell }(b),1/\tilde{q}^{\ell }(c)\}\). From (25), (44), (45) and (46) we thus obtain

$$\begin{aligned} 0= \int \limits _{b}^{c}\left( \frac{\partial G}{\partial x}(b,x)- \frac{\partial G}{\partial x}(c,x)\right) h(x)\,\mathrm{d}x \!\ge \! C\int \limits _{b}^{c} (g_{a}(x)+\mu g_{b}(x))\,\mathrm{d}x \!=\! C\int \limits _{b}^{c}g_{a}(x)\,\mathrm{d}x\!>\!0, \end{aligned}$$

which is a contradiction, and the lemma follows. \(\square \)

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Bruno, O.P., Prieto, A. Spatially Dispersionless, Unconditionally Stable FC–AD Solvers for Variable-Coefficient PDEs. J Sci Comput 58, 331–366 (2014). https://doi.org/10.1007/s10915-013-9734-8

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