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**Implicit, compact, linearized \(\theta\)-methods with factorization for multidimensional reaction-diffusion equations.**
*(English)*
Zbl 0943.65098

Summary: An iterative predictor-corrector technique for the elimination of the approximate factorization errors which result from the factorization of implicit, three-point compact, linearized \(\theta\)-methods in multidimensional reaction-diffusion equations is proposed, and its convergence and linear stability are analyzed. Four compact, approximate factorization techniques which do not account for the approximate factorization errors and which involve three-point stencils for each one-dimensional operator are developed.

The first technique uses the full Jacobian matrix of the reaction terms, requires the inversion of, in general, dense matrices, and its approximate factorization errors are second-order accurate in time. The second and third methods approximate the Jacobian matrix by diagonal or triangular ones which are easily inverted but their approximate factorization errors are, however, first-order accurate in time. The fourth approximately factorized, compact, implicit method has approximate factorization errors which are second-order accurate in time and requires the inversion of lower and upper triangular matrices.

The techniques are applied to a nonlinear, two-species, two-dimensional system of reaction-diffusion equations in order to determine the approximate factorization errors and those resulting from the approximations to the Jacobian matrix as functions of the allocation of the reaction terms, space and time.

The first technique uses the full Jacobian matrix of the reaction terms, requires the inversion of, in general, dense matrices, and its approximate factorization errors are second-order accurate in time. The second and third methods approximate the Jacobian matrix by diagonal or triangular ones which are easily inverted but their approximate factorization errors are, however, first-order accurate in time. The fourth approximately factorized, compact, implicit method has approximate factorization errors which are second-order accurate in time and requires the inversion of lower and upper triangular matrices.

The techniques are applied to a nonlinear, two-species, two-dimensional system of reaction-diffusion equations in order to determine the approximate factorization errors and those resulting from the approximations to the Jacobian matrix as functions of the allocation of the reaction terms, space and time.

### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

35K57 | Reaction-diffusion equations |