A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations (Communications in Computational Physics, 2020)
In this paper, a class of high order numerical schemes is proposed to solvethe nonlinear parabolic equations with variable coefficients. This method is based onour previous work [11] for convection-diffusion equations, which relies on a specialkernel-based formulation of the solutions and successive convolution. However, disadvantages appear when we extend the previous method to our equations, such as inefficient choice of parameters and unprovable stability for high-dimensional problems.To overcome these difficulties, a new kernel-based formulation is designed to approachthe spatial derivatives. It maintains the good properties of the original one, including the high order accuracy and unconditionally stable for one-dimensional problems,hence allowing much larger time step evolution compared with other explicit schemes.In addition, without extra computational cost, the proposed scheme can enlarge theavailable interval of the special parameter in the formulation, leading to less errorsand higher efficiency. Moreover, theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well. We present numerical tests forone- and two-dimensional scalar and system, demonstrating the designed high orderaccuracy and unconditionally stable property of the scheme.
Keywords: Nonlinear parabolic equation; kernel based scheme; unconditionally stable; high order accuracy.