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Kernel Based High Order “Explicit” Unconditionally Stable Scheme for Nonlinear Degenerate Advection-Diffusion Equations

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Abstract

In this paper, we present a novel numerical scheme for solving a class of nonlinear degenerate parabolic equations with non-smooth solutions. The proposed method relies on a special kernel based formulation of the solutions found in our early work on the method of lines transpose and successive convolution. In such a framework, a high order weighted essentially non-oscillatory methodology and a nonlinear filter are further employed to avoid spurious oscillations. High order accuracy in time is realized by using the high order explicit strong-stability-preserving (SSP) Runge-Kutta method. Moreover, theoretical investigations of the kernel based formulation combined with an explicit SSP method indicate that the combined scheme is unconditionally stable and up to third order accuracy. Evaluation of the kernel based approach is done with a fast \({\mathcal {O}}(N)\) summation algorithm. The new method allows for much larger time step evolution compared with other explicit schemes with the same order accuracy, leading to remarkable computational savings.

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References

  1. 1.

    Abedian, R., Adibi, H., Dehghan, M.: A high-order weighted essentially non-oscillatory (WENO) finite difference scheme for nonlinear degenerate parabolic equations. Comput. Phys. Commun. 184(8), 1874–1888 (2013)

  2. 2.

    Alt, H.W., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Mathematische Zeitschrift 183(3), 311–341 (1983)

  3. 3.

    Aregba-Driollet, D., Natalini, R., Tang, S.: Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems. Math. Comput. 73(245), 63–94 (2004)

  4. 4.

    Aronson, D.G.: The Porous Medium Equation. Nonlinear Diffusion Problems, pp. 1–46. Springer, Berlin (1986)

  5. 5.

    Ascher, U., Ruuth, S., Spiteri, R.: Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25(2–3), 151–167 (1997)

  6. 6.

    Barenblatt, G.I.: On self-similar motions of a compressible fluid in a porous medium. Akad. Nauk SSSR. Prikl. Mat. Meh 16(6), 79–6 (1952)

  7. 7.

    Barnes, J., Hut, P.: A hierarchical \(O(N \log N)\) force-calculation algorithm. Nature 324, 446–449 (1986)

  8. 8.

    Bessemoulin-Chatard, M., Filbet, F.: A finite volume scheme for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput. 34(5), B559–B583 (2012)

  9. 9.

    Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227(6), 3191–3211 (2008)

  10. 10.

    Buckley, S.E., Leverett, M., et al.: Mechanism of fluid displacement in sands. Trans. AIME 146(01), 107–116 (1942)

  11. 11.

    Causley, M., Cho, H., Christlieb, A.: Method of lines transpose: energy gradient flows using direct operator inversion for phase field models (2016). arXiv preprint arXiv:1611.04214

  12. 12.

    Causley, M., Cho, H., Christlieb, A., Seal, D.: Method of lines transpose: High order L-stable \({\cal{O}}(N)\) schemes for parabolic equations using successive convolution. SIAM J. Numer. Anal. 54(3), 1635–1652 (2016)

  13. 13.

    Causley, M., Christlieb, A., Ong, B., Van Groningen, L.: Method of lines transpose: an implicit solution to the wave equation. Math. Comput. 83(290), 2763–2786 (2014)

  14. 14.

    Causley, M.F., Christlieb, A.J.: Higher order A-stable schemes for the wave equation using a successive convolution approach. SIAM J. Numer. Anal. 52(1), 220–235 (2014)

  15. 15.

    Causley, M.F., Christlieb, A.J., Guclu, Y., Wolf, E.: Method of lines transpose: a fast implicit wave propagator (2013). arXiv preprint arXiv:1306.6902

  16. 16.

    Cavalli, F., Naldi, G., Puppo, G., Semplice, M.: High-order relaxation schemes for nonlinear degenerate diffusion problems. SIAM J. Numer. Anal. 45(5), 2098–2119 (2007)

  17. 17.

    Cheng, Y., Christlieb, A.J., Guo, W., Ong, B.: An asymptotic preserving Maxwell solver resulting in the Darwin limit of electrodynamics. J. Sci. Comput. 71, 1–35 (2015)

  18. 18.

    Chou, C.-S., Shu, C.-W.: High order residual distribution conservative finite difference WENO schemes for steady state problems on non-smooth meshes. J. Comput. Phys. 214(2), 698–724 (2006)

  19. 19.

    Chou, C.-S., Shu, C.-W.: High order residual distribution conservative finite difference WENO schemes for convection-diffusion steady state problems on non-smooth meshes. J. Comput. Phys. 224(2), 992–1020 (2007)

  20. 20.

    Christlieb, A., Guo, W., Jiang, Y.: A WENO-based Method of Lines Transpose approach for Vlasov simulations. J. Comput. Phys. 327, 337–367 (2016)

  21. 21.

    Christlieb, A., Guo, W., Jiang, Y.: A kernel based high order “explicit” unconditionally stable scheme for time dependent Hamilton-Jacobi equations. J. Comput. Phys. 379, 214–236 (2019)

  22. 22.

    Christlieb, A., Guo, W., Jiang, Y., Yang, H.: https://github.com/hyoseonyang/molt-tutorial.git (2019)

  23. 23.

    Duyn, Cv, Peletier, L.: Nonstationary filtration in partially saturated porous media. Arch. Ration. Mech. Anal. 78(2), 173–198 (1982)

  24. 24.

    Gottlieb, S.: On high order strong stability preserving Runge-Kutta and multi step time discretizations. J. Sci. Comput. 25(1), 105–128 (2005)

  25. 25.

    Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)

  26. 26.

    Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987)

  27. 27.

    Gustafsson, B., Kreiss, H.-O., Sundström, A.: Stability theory of difference approximations for mixed initial boundary value problems. II. Mathematics of Computation, pp. 649–686, (1972)

  28. 28.

    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems, 2nd edn. Springer, Berlin (1996)

  29. 29.

    Hajipour, M., Malek, A.: High accurate NRK and MWENO scheme for nonlinear degenerate parabolic PDEs. Appl. Math. Modell. 36(9), 4439–4451 (2012)

  30. 30.

    Jia, J., Huang, J.: Krylov deferred correction accelerated method of lines transpose for parabolic problems. J. Comput. Phys. 227(3), 1739–1753 (2008)

  31. 31.

    Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted eno schemes. J. Comput. Phys. 126(1), 202–228 (1996)

  32. 32.

    Ketcheson, D.I.: Step sizes for strong stability preservation with downwind-biased operators. SIAM J. Numer. Anal. 49(4), 1649–1660 (2011)

  33. 33.

    Kropinski, M.C.A., Quaife, B.D.: Fast integral equation methods for Rothe’s method applied to the isotropic heat equation. Comput. Math. Appl. 61(9), 2436–2446 (2011)

  34. 34.

    Kurganov, A., Tadmor, E.: New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160(1), 241–282 (2000)

  35. 35.

    Liu, Y., Shu, C.-W., Zhang, M.: On the positivity of linear weights in WENO approximations. Acta Math. Appl. Sin. Engl. Ser. 25(3), 503–538 (2009)

  36. 36.

    Liu, Y., Shu, C.-W., Zhang, M.: High order finite difference WENO schemes for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput. 33(2), 939–965 (2011)

  37. 37.

    Muskat, M.: The flow of homogeneous fluids through porous media. Soil Sci. 46(2), 169 (1938)

  38. 38.

    Otto, F.: L1-contraction and uniqueness for quasilinear elliptic-parabolic equations. J. Diff. Equ. 131(1), 20–38 (1996)

  39. 39.

    Salazar, A., Raydan, M., Campo, A.: Theoretical analysis of the exponential transversal method of lines for the diffusion equation. Numer. Methods Partial Diff. Equ. 16(1), 30–41 (2000)

  40. 40.

    Schemann, M., Bornemann, F.A.: An adaptive rothe method for the wave equation. Comput. Vis. Sci. 1(3), 137–144 (1998)

  41. 41.

    Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, pp. 325–432. Springer (1998)

  42. 42.

    Shu, C.-W.: A survey of strong stability preserving high order time discretizations. Collected Lectures Preserv. Stab. Discret. 109, 51–65 (2002)

  43. 43.

    Shu, C.-W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51(1), 82–126 (2009)

  44. 44.

    Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)

  45. 45.

    Zel’dovich, Y.B., Kompaneets, A.: Towards a theory of heat conduction with thermal conductivity depending on the temperature. In: Collection of Papers Dedicated to 70th Birthday of Academician AF Ioffe, Izd. Akad. Nauk SSSR, Moscow, pp. 61–71 (1950)

  46. 46.

    Zhang, Q., Wu, Z.-L.: Numerical simulation for porous medium equation by local discontinuous galerkin finite element method. J. Sci. Comput. 38(2), 127–148 (2009)

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Correspondence to Yan Jiang.

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Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A. Christlieb: Research is supported in part by AFOSR grants FA9550-12-1-0343, FA9550-12-1-0455, and FA9550-15-1-0282, and NSF Grant DMS-1418804. W. Guo: Research is supported in part by NSF Grants DMS-1620047,DMS-1830838. Y. Jiang: Research is supported in part by NSFC Grant 11901555.

Appendix A. Proof of Lemma 3.2

Appendix A. Proof of Lemma 3.2

Here, we only give the proof for the case of \({\mathcal {D}}_{0}\) and \({\mathcal {D}}_{L}\). For \({\mathcal {D}}_{R}\), the proof can be established by a similar idea.

Using the definition of \(I^{0}\) and integration by parts twice, we have

$$\begin{aligned} I^{0}[v,\alpha ](x)&=v(x) + \frac{1}{\alpha ^2}I^{0}[v_{xx},\alpha ](x) - \left( \frac{1}{2}v(a)-\frac{1}{2\alpha }v_{x}(a)\right) e^{-\alpha (x-a)}\nonumber \\&-\left( \frac{1}{2}v(b)+\frac{1}{2\alpha }v_{x}(b)\right) e^{-\alpha (b-x)}. \end{aligned}$$
(A.1)

Thus,

$$\begin{aligned} {\mathcal {D}}_{0}[v,\alpha ](x) =&-\frac{1}{\alpha ^{2}}I^{0}[v_{xx},\alpha ](x) -\left( A_{0}[v,\alpha ]-\frac{1}{2}v(a)+\frac{1}{2\alpha }v_{x}(a)\right) e^{-\alpha (x-a)}\\&-\left( B_{0}[v,\alpha ]-\frac{1}{2}v(b)-\frac{1}{2\alpha }v_{x}(b)\right) e^{-\alpha (b-x)}. \end{aligned}$$

Here, \(A_{0}[v,\alpha ]\) and \(B_{0}[v,\alpha ]\) are obtained from the boundary treatment of \({\mathcal {D}}_{0}[v,\alpha ]\) (2.5). Moreover, based on (A.1), \(A_{0}[v,\alpha ]\) and \(B_{0}[v,\alpha ]\) can be rewritten as

$$\begin{aligned} A_{0}[v,\alpha ]=&\frac{1}{1-\mu } \left( \frac{1}{\alpha ^2} I^{0}[v_{xx},\alpha ](b) - \frac{1}{2} \left( v(a)-\frac{1}{\alpha }v_{x}(a) \right) \mu + \frac{1}{2}\left( v(b)-\frac{1}{\alpha }v_{x}(b) \right) \right) ,\\ B_{0}[v,\alpha ]=&\frac{1}{1-\mu } \left( \frac{1}{\alpha ^2} I^{0}[v_{xx},\alpha ](a) + \frac{1}{2}\left( v(a)+\frac{1}{\alpha }v_{x}(a) \right) - \frac{1}{2}\left( v(b)+\frac{1}{\alpha }v_{x}(b) \right) \mu \right) . \end{aligned}$$

Therefore, we have

$$\begin{aligned} {\mathcal {D}}_{0}[v,\alpha ](x) =&-\frac{1}{\alpha ^{2}}I^{0}[v_{xx},\alpha ](x) -\frac{1}{\alpha ^2}\frac{I^{0}[v_{xx},\alpha ](b)}{1-\mu }e^{-\alpha (x-a)} -\frac{1}{\alpha ^2}\frac{I^{0}[v_{xx},\alpha ](a)}{1-\mu } e^{\alpha (b-x)}\\ =&-\frac{1}{\alpha ^2} {\mathcal {L}}_{0}^{-1}[v_{xx},\alpha ](x) = -\frac{1}{\alpha ^2} v_{xx}(x) + \frac{1}{\alpha ^2}{\mathcal {D}}_{0}[v_{xx},\alpha ](x) \end{aligned}$$

Upon iterating this process k times, we obtain that

$$\begin{aligned} {\mathcal {D}}_{0}[v,\alpha ](x) =&-\sum _{p=1}^{k}\frac{1}{\alpha ^{2p}}\partial ^{2p}_{x}v(x) -\frac{1}{\alpha ^{2k+2}}{\mathcal {L}}^{-1}_{0}[\partial ^{2k+2}_{x}v,\alpha ](x). \end{aligned}$$

Similarly, we can do integration by parts on \(I^{L}\), and have

$$\begin{aligned} I^{L}[v,\alpha ] =&-\frac{1}{\alpha }I^{L}[v_{x},\alpha ](x) + v(x) - v(a)e^{-\alpha (x-a)}, \\ A_{L}[v,\alpha ] =&-\frac{1}{\alpha }A_{L}[v_{x},\alpha ] + \frac{1}{1-\mu }\left( v(b) -v(a)e^{-\alpha (b-a)} \right) = -\frac{1}{\alpha }A_{L}[v_{x},\alpha ] + v(a). \end{aligned}$$

Therefore, we have

$$\begin{aligned} {\mathcal {D}}_{L}[v,\alpha ]=&\frac{1}{\alpha } I^{L}[v_x,\alpha ](x) + \frac{1}{\alpha } A_{L}[v_x,\alpha ] e^{-\alpha (x-a)} =\frac{1}{\alpha } {\mathcal {L}}_{L}^{-1}[v_{x},\alpha ](x) \\ =&\frac{1}{\alpha } v_{x}(x) - \frac{1}{\alpha }{\mathcal {D}}_{L}[v_x,\alpha ](x)\\ =&-\sum _{p=1}^{k}\left( -\frac{1}{\alpha }\right) ^p \partial _{x}^pv(x) + \left( -\frac{1}{\alpha }\right) {\mathcal {L}}^{-1}_{L}[\partial _{x}^{k+1}v,\alpha ](x), \end{aligned}$$

and the lemma is proved.

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Christlieb, A., Guo, W., Jiang, Y. et al. Kernel Based High Order “Explicit” Unconditionally Stable Scheme for Nonlinear Degenerate Advection-Diffusion Equations. J Sci Comput 82, 52 (2020). https://doi.org/10.1007/s10915-020-01152-w

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Keywords

  • Integral solution
  • Unconditionally stable
  • Weighted essentially non-oscillatory methodology
  • High order accuracy
  • Nonlinear degenerate advection-diffusion equation