TY - JOUR
T1 - Kernel Based High Order “Explicit” Unconditionally Stable Scheme for Nonlinear Degenerate Advection-Diffusion Equations
AU - Christlieb, Andrew
AU - Guo, Wei
AU - Jiang, Yan
AU - Yang, Hyoseon
N1 - Funding Information:
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.
Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2020/3/1
Y1 - 2020/3/1
N2 - 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 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.
AB - 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 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.
KW - High order accuracy
KW - Integral solution
KW - Nonlinear degenerate advection-diffusion equation
KW - Unconditionally stable
KW - Weighted essentially non-oscillatory methodology
UR - http://www.scopus.com/inward/record.url?scp=85079572729&partnerID=8YFLogxK
U2 - 10.1007/s10915-020-01152-w
DO - 10.1007/s10915-020-01152-w
M3 - Article
AN - SCOPUS:85079572729
VL - 82
JO - Journal of Scientific Computing
JF - Journal of Scientific Computing
SN - 0885-7474
IS - 3
M1 - 52
ER -