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 [11] 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.
A new type hybrid Hermite weighted essentially non-oscillatory ( HWENO) schemes in the implicit method of lines transpose ( MOL T ) framework is designed for solving one-dimensional linear transport equations and further applied to the Vlasov-Poisson (VP) simulations via dimensional splitting. Compared with the WENO-based MOL T method given in J. Comput. Phys. [2016, 327: 337-367], the new proposed hybrid HWENO-based MOL T scheme has two advantages. The first is the HWENO schemes using the stencils narrower than those of the WENO schemes with the same order of accuracy. The second is that the schemes can adapt between the linear scheme and the HWENO scheme automatically. In summary, the hybrid HWENO scheme keeps the simplicity and robustness of the simple WENO scheme, while it has higher efficiency with less numerical errors in smooth regions and less computational costs as well. Benchmark examples are given to demonstrate the robustness and good performance of the proposed scheme.
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