  In this paper, a class of high order numerical schemes is proposed to solve
the nonlinear parabolic equations with variable coefficients. This method is
based on our previous work [10] for convection-diffusion equations, which
relies on a special kernel-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 approach the 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 additional, without extra computational cost, the proposed scheme can
enlarge the available interval of the special parameter in the formulation,
leading to less errors and higher efficiency. Moreover, theoretical
investigations indicate that it is unconditionally stable for multi-dimensional
problems as well. We present numerical tests for one- and two-dimensional
scalar and system, demonstrating the designed high order accuracy and
unconditionally stable property of the scheme.
