Hermitian operator methods for reaction‐diffusion equations

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Abstract
A variety of time‐linearization, quasilinearization, operator‐splitting, and implicit techniques which use compact or Hermitian operators has been developed for and applied to one‐dimensional reaction‐diffusion equations. Compact operators are compared with second‐order accurate spatial approximations in order to assess the accuracy and efficiency of Hermitian techniques. It is shown that time‐linearization, quasilinearization, and implicit techniques which use compact operators are less accurate than second‐order accurate spatial discretizations if first‐order approximations are employed to evaluate the time derivatives. This is attributed to first‐order accurati temporal truncation errors. Compact operator techniques which use second‐order temporal approximations are found to be more accurate and efficient than second‐order accurate, in both space and time, algorithms. Quasilinearization methods are found to be more accurate than time‐linearization schemes. However, quasilinearization techniques are less efficient because they require the inversion of block tridiagonal matrices at each iteration. Some improvements in accuracy can be obtained by using partial quasilinearization and linearizing each equation with respect to the variable whose equation is being solved. Operator‐splitting methods which use compact differences to evaluate the diffusion operator were found to be less accurate than operator‐splitting procedures employ second‐order accurate spatial approximations. Comparisons among the methods presented in this paper are shown in terms of the L2‐norm errors and computed wave speeds for a variety of time steps and grid spacings: The numerical efficiency is assessed in terms of the CPU time required to achieve the same accuracy.

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