In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial deriv...In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.展开更多
In this work,we construct two efficient fully decoupled,linear,unconditionally stable numerical algorithms for the thermally coupled incompressible magnetohydrodynamic equations.Firstly,in order to obtain the desired ...In this work,we construct two efficient fully decoupled,linear,unconditionally stable numerical algorithms for the thermally coupled incompressible magnetohydrodynamic equations.Firstly,in order to obtain the desired algorithm,we introduce a scalar auxiliary variable(SAV)to get a new equivalent system.Secondly,by combining the pressure-correction method and the explicit-implicit method,we perform semi-discrete numerical algorithms of first and second order,respectively.Then,we prove that the obtained algorithms follow an unconditionally stable law in energy,and we provide a detailed implementation process,which we only need to solve a series of linear differential equations with constant coefficients at each time step.More importantly,with some powerful analysis,we give the order of convergence of the errors.Finally,to illustrate theoretical results,some numerical experiments are given.展开更多
We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation.Due to the impact of the variable diffusivity coefficient,ex...We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation.Due to the impact of the variable diffusivity coefficient,existing regularity results for their constantcoefficient counterparts do not apply,while the bilinear forms of the state(adjoint)equation may lose the coercivity that is critical in error estimates of the finite element method.We reformulate the state equation as an equivalent constant-coefficient fractional diffusion equation with the addition of a variable-coefficient low-order fractional advection term.First order optimality conditions are accordingly derived and the smoothing properties of the solutions are analyzed by,e.g.,interpolation estimates.The weak coercivity of the resulting bilinear forms are proven via the Garding inequality,based on which we prove the optimal-order convergence estimates of the finite element method for the(adjoint)state variable and the control variable.Numerical experiments substantiate the theoretical predictions.展开更多
Forest volume, the major component of forest biomass, is an important issue in forest resource monitoring.It is estimated from tree volume tables or equations. Based on tree volume data of 1840 sample trees from Chine...Forest volume, the major component of forest biomass, is an important issue in forest resource monitoring.It is estimated from tree volume tables or equations. Based on tree volume data of 1840 sample trees from Chinese fir （Cunninghamia lanceolata） plantations in Guizhou Province in southwestern China, parallel one- and two-variable tree volume tables and tree height curves for central and other areas were constructed using an error-in-variable modeling method. The results show that, although the one-variable tree volume equations and height curves between the central and other areas were significantly different, the two-variable volume equations were sufficiently close, so that a generalized two-variable tree volume equation could be established for the entire province.展开更多
In this paper,a linearized energy-stable scalar auxiliary variable(SAV)Galerkin scheme is investigated for a two-dimensional nonlinear wave equation and the unconditional superconvergence error estimates are obtained ...In this paper,a linearized energy-stable scalar auxiliary variable(SAV)Galerkin scheme is investigated for a two-dimensional nonlinear wave equation and the unconditional superconvergence error estimates are obtained without any certain time-step restrictions.The key to the analysis is to derive the boundedness of the numerical solution in the H^(1)-norm,which is different from the temporal-spatial error splitting approach used in the previous literature.Meanwhile,numerical results are provided to confirm the theoretical findings.展开更多
文摘In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.
基金Supported by Research Project Supported by Shanxi Scholarship Council of China(2021-029)Shanxi Provincial International Cooperation Base and Platform Project(202104041101019)Shanxi Province Natural Science Foundation(202203021211129)。
文摘In this work,we construct two efficient fully decoupled,linear,unconditionally stable numerical algorithms for the thermally coupled incompressible magnetohydrodynamic equations.Firstly,in order to obtain the desired algorithm,we introduce a scalar auxiliary variable(SAV)to get a new equivalent system.Secondly,by combining the pressure-correction method and the explicit-implicit method,we perform semi-discrete numerical algorithms of first and second order,respectively.Then,we prove that the obtained algorithms follow an unconditionally stable law in energy,and we provide a detailed implementation process,which we only need to solve a series of linear differential equations with constant coefficients at each time step.More importantly,with some powerful analysis,we give the order of convergence of the errors.Finally,to illustrate theoretical results,some numerical experiments are given.
基金supported by the National Natural Science Foundation of China(11971276,12171287)Natural Science Foundation of Shandong Province(ZR2016JL004)+1 种基金supported by the China Postdoctoral Science Foundation(2021TQ0017,2021M700244)International Postdoctoral Exchange Fellowship Program(Talent-Introduction Program)(YJ20210019)。
文摘We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation.Due to the impact of the variable diffusivity coefficient,existing regularity results for their constantcoefficient counterparts do not apply,while the bilinear forms of the state(adjoint)equation may lose the coercivity that is critical in error estimates of the finite element method.We reformulate the state equation as an equivalent constant-coefficient fractional diffusion equation with the addition of a variable-coefficient low-order fractional advection term.First order optimality conditions are accordingly derived and the smoothing properties of the solutions are analyzed by,e.g.,interpolation estimates.The weak coercivity of the resulting bilinear forms are proven via the Garding inequality,based on which we prove the optimal-order convergence estimates of the finite element method for the(adjoint)state variable and the control variable.Numerical experiments substantiate the theoretical predictions.
基金supported by the Agricultural Science and Technique Foundation of Guizhou Province, China (No. 2008-3059)the Research Funds of Forestry Administration of Guizhou Province, China (Nos. 2010-01-08, 2010-01, 200625)
文摘Forest volume, the major component of forest biomass, is an important issue in forest resource monitoring.It is estimated from tree volume tables or equations. Based on tree volume data of 1840 sample trees from Chinese fir （Cunninghamia lanceolata） plantations in Guizhou Province in southwestern China, parallel one- and two-variable tree volume tables and tree height curves for central and other areas were constructed using an error-in-variable modeling method. The results show that, although the one-variable tree volume equations and height curves between the central and other areas were significantly different, the two-variable volume equations were sufficiently close, so that a generalized two-variable tree volume equation could be established for the entire province.
基金supported by the National Natural Science Foundation of China(No.12101568).
文摘In this paper,a linearized energy-stable scalar auxiliary variable(SAV)Galerkin scheme is investigated for a two-dimensional nonlinear wave equation and the unconditional superconvergence error estimates are obtained without any certain time-step restrictions.The key to the analysis is to derive the boundedness of the numerical solution in the H^(1)-norm,which is different from the temporal-spatial error splitting approach used in the previous literature.Meanwhile,numerical results are provided to confirm the theoretical findings.
