In this paper, we implement energy equation coupled with viscous Burgers’ equation as a mathematical model for the estimation of thermal pollution of river water. The model is a nonlinear system of partial differenti...In this paper, we implement energy equation coupled with viscous Burgers’ equation as a mathematical model for the estimation of thermal pollution of river water. The model is a nonlinear system of partial differential equations (PDEs) that read as an initial and boundary value problem (IBVP). For the numerical solution of the IBVP, we investigate an explicit second-order Lax- Wendroff type scheme for nonlinear parabolic PDEs. We present the numerical solutions graphically as a temperature profile, which shows good qualitative agreement with natural phenomena of heat transfer. We estimate the thermal pollution of water caused by industrialization on the bank of a river.展开更多
The incompatibilities between the initial and boundary data will cause singularities at the time-space corners,which in turn adversely affect the accuracy of the numerical schemes used to compute the solutions.We stud...The incompatibilities between the initial and boundary data will cause singularities at the time-space corners,which in turn adversely affect the accuracy of the numerical schemes used to compute the solutions.We study the corner singularity issue for nonlinear evolution equations in 1D,and propose two remedy procedures that effectively recover much of the accuracy of the numerical scheme in use.Applications of the remedy procedures to the 1D viscous Burgers equation,and to the 1D nonlinear reaction-diffusion equation are presented.The remedy procedures are applicable to other nonlinear diffusion equations as well.展开更多
文摘In this paper, we implement energy equation coupled with viscous Burgers’ equation as a mathematical model for the estimation of thermal pollution of river water. The model is a nonlinear system of partial differential equations (PDEs) that read as an initial and boundary value problem (IBVP). For the numerical solution of the IBVP, we investigate an explicit second-order Lax- Wendroff type scheme for nonlinear parabolic PDEs. We present the numerical solutions graphically as a temperature profile, which shows good qualitative agreement with natural phenomena of heat transfer. We estimate the thermal pollution of water caused by industrialization on the bank of a river.
基金supported in part by NSF grants DMS0604235 and DMS0906440the Research Fund of Indiana University.
文摘The incompatibilities between the initial and boundary data will cause singularities at the time-space corners,which in turn adversely affect the accuracy of the numerical schemes used to compute the solutions.We study the corner singularity issue for nonlinear evolution equations in 1D,and propose two remedy procedures that effectively recover much of the accuracy of the numerical scheme in use.Applications of the remedy procedures to the 1D viscous Burgers equation,and to the 1D nonlinear reaction-diffusion equation are presented.The remedy procedures are applicable to other nonlinear diffusion equations as well.
