Transient diffusion equations arise in many branches of engineering and applied sciences(e.g.,heat transfer and mass transfer),and are parabolic partial differential equations.It is well-known that these equations sat...Transient diffusion equations arise in many branches of engineering and applied sciences(e.g.,heat transfer and mass transfer),and are parabolic partial differential equations.It is well-known that these equations satisfy important mathematical properties like maximum principles and the non-negative constraint,which have implications in mathematical modeling.However,existing numerical formulations for these types of equations do not,in general,satisfy maximum principles and the nonnegative constraint.In this paper,we present a methodology for enforcing maximum principles and the non-negative constraint for transient anisotropic diffusion equation.The proposed methodology is based on the method of horizontal lines in which the time is discretized first.This results in solving steady anisotropic diffusion equation with decay equation at every discrete time-level.We also present other plausible temporal discretizations,and illustrate their shortcomings in meeting maximum principles and the non-negative constraint.The proposedmethodology can handle general computational grids with no additional restrictions on the time-step.We illustrate the performance and accuracy of the proposed methodology using representative numerical examples.We also perform a numerical convergence analysis of the proposed methodology.For comparison,we also present the results from the standard singlefield semi-discrete formulation and the results froma popular software package,which all will violate maximum principles and the non-negative constraint.展开更多
基金K.B.N.and M.S.acknowledge the support from the National Science Foundation under GrantNo.CMMI 1068181.K.B.N.also acknowledges the supports fromtheDOE Office of Nuclear Energy’s Nuclear Energy University Programs(NEUP)The opinions expressed in this paper are those of the authors and do not necessarily reflect that of the sponsors。
文摘Transient diffusion equations arise in many branches of engineering and applied sciences(e.g.,heat transfer and mass transfer),and are parabolic partial differential equations.It is well-known that these equations satisfy important mathematical properties like maximum principles and the non-negative constraint,which have implications in mathematical modeling.However,existing numerical formulations for these types of equations do not,in general,satisfy maximum principles and the nonnegative constraint.In this paper,we present a methodology for enforcing maximum principles and the non-negative constraint for transient anisotropic diffusion equation.The proposed methodology is based on the method of horizontal lines in which the time is discretized first.This results in solving steady anisotropic diffusion equation with decay equation at every discrete time-level.We also present other plausible temporal discretizations,and illustrate their shortcomings in meeting maximum principles and the non-negative constraint.The proposedmethodology can handle general computational grids with no additional restrictions on the time-step.We illustrate the performance and accuracy of the proposed methodology using representative numerical examples.We also perform a numerical convergence analysis of the proposed methodology.For comparison,we also present the results from the standard singlefield semi-discrete formulation and the results froma popular software package,which all will violate maximum principles and the non-negative constraint.
