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

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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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Correspondence to Yan Jiang.

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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

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


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

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).

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