In this paper,we consider numerical methods for two-sided space variable-order fractional diffusion equations(VOFDEs)with a nonlinear source term.The implicit Euler(IE)method and a shifted Grünwald(SG)scheme are ...In this paper,we consider numerical methods for two-sided space variable-order fractional diffusion equations(VOFDEs)with a nonlinear source term.The implicit Euler(IE)method and a shifted Grünwald(SG)scheme are used to approximate the temporal derivative and the space variable-order(VO)fractional derivatives,respectively,which leads to an IE-SG scheme.Since the order of the VO derivatives depends on the space and the time variables,the corresponding coefficient matrices arising from the discretization of VOFDEs are dense and without the Toeplitz-like structure.In light of the off-diagonal decay property of the coefficient matrices,we consider applying the preconditioned generalized minimum residual methods with banded preconditioners to solve the discretization systems.The eigenvalue distribution and the condition number of the preconditioned matrices are studied.Numerical results show that the proposed banded preconditioners are efficient.展开更多
文摘In this paper,we consider numerical methods for two-sided space variable-order fractional diffusion equations(VOFDEs)with a nonlinear source term.The implicit Euler(IE)method and a shifted Grünwald(SG)scheme are used to approximate the temporal derivative and the space variable-order(VO)fractional derivatives,respectively,which leads to an IE-SG scheme.Since the order of the VO derivatives depends on the space and the time variables,the corresponding coefficient matrices arising from the discretization of VOFDEs are dense and without the Toeplitz-like structure.In light of the off-diagonal decay property of the coefficient matrices,we consider applying the preconditioned generalized minimum residual methods with banded preconditioners to solve the discretization systems.The eigenvalue distribution and the condition number of the preconditioned matrices are studied.Numerical results show that the proposed banded preconditioners are efficient.
