In 1992, Cooper [2] has presented some new stability concepts for Runge-Kutta methods whichis based on two slightly different test problems, and obtained the algebraic conditions that guarantee newstability properties...In 1992, Cooper [2] has presented some new stability concepts for Runge-Kutta methods whichis based on two slightly different test problems, and obtained the algebraic conditions that guarantee newstability properties. In this paper, we extend these results to general linear methods and to more generalproblem class Kστ. The concepts of (k, p, q)-secondary stability and (k, p. q)-secondary stability are introduced, and the criteria of secondary algebraic stability are also established. The criteria relax algebraicstability conditions while retaining the virtues of a nonlinear test problem.展开更多
The exponentially expanded space grid was incorporated into the network approach to overcome the problem of low simulation efficiency during the simulations of electrochemical problems with stiff kinetics or wide disp...The exponentially expanded space grid was incorporated into the network approach to overcome the problem of low simulation efficiency during the simulations of electrochemical problems with stiff kinetics or wide dispersion of diffusion coefficients, resulting in an effective electrochemical simulation method: exponentially expanded grid network approach (EEGNA). The stability and accuracy of the EEGNA for the simulation of various electrode processes coupled with different types of homogeneous reactions were investigated.展开更多
We develop an efficient and accurate spectral deferred correction(SDC)method for fractional differential equations(FDEs)by extending the algorithm in[14]for classical ordinary differential equations(ODEs).Specifically...We develop an efficient and accurate spectral deferred correction(SDC)method for fractional differential equations(FDEs)by extending the algorithm in[14]for classical ordinary differential equations(ODEs).Specifically,we discretize the resulted Picard integral equation by the SDC method and accelerate the convergence of the SDC iteration by using the generalized minimal residual algorithm(GMRES).We first derive the correction matrix of the SDC method for FDEs and analyze the convergence region of the SDC method.We then present several numerical examples for stiff and non-stiff FDEs including fractional linear and nonlinear ODEs as well as fractional phase field models,demonstrating that the accelerated SDC method is much more efficient than the original SDC method,especially for stiff problems.Furthermore,we resolve the issue of low accuracy arising from the singularity of the solutions by using a geometric mesh,leading to highly accurate solutions compared to uniform mesh solutions at almost the same computational cost.Moreover,for long-time integration of FDEs,using the geometric mesh leads to great computational savings as the total number of degrees of freedom required is relatively small.展开更多
文摘In 1992, Cooper [2] has presented some new stability concepts for Runge-Kutta methods whichis based on two slightly different test problems, and obtained the algebraic conditions that guarantee newstability properties. In this paper, we extend these results to general linear methods and to more generalproblem class Kστ. The concepts of (k, p, q)-secondary stability and (k, p. q)-secondary stability are introduced, and the criteria of secondary algebraic stability are also established. The criteria relax algebraicstability conditions while retaining the virtues of a nonlinear test problem.
文摘The exponentially expanded space grid was incorporated into the network approach to overcome the problem of low simulation efficiency during the simulations of electrochemical problems with stiff kinetics or wide dispersion of diffusion coefficients, resulting in an effective electrochemical simulation method: exponentially expanded grid network approach (EEGNA). The stability and accuracy of the EEGNA for the simulation of various electrode processes coupled with different types of homogeneous reactions were investigated.
基金Z.Mao was supported by the Fundamental Research Funds for the Central Universities(Grant 20720210037)G.E.Karniadakis was supported by the MURI/ARO on Fractional PDEs for Conservation Laws and Beyond:Theory,Numerics and Applications(Grant W911NF-15-1-0562)X.Chen was supported by the Fujian Provincial Natural Science Foundation of China(Grants 2022J01338,2020J01703).
文摘We develop an efficient and accurate spectral deferred correction(SDC)method for fractional differential equations(FDEs)by extending the algorithm in[14]for classical ordinary differential equations(ODEs).Specifically,we discretize the resulted Picard integral equation by the SDC method and accelerate the convergence of the SDC iteration by using the generalized minimal residual algorithm(GMRES).We first derive the correction matrix of the SDC method for FDEs and analyze the convergence region of the SDC method.We then present several numerical examples for stiff and non-stiff FDEs including fractional linear and nonlinear ODEs as well as fractional phase field models,demonstrating that the accelerated SDC method is much more efficient than the original SDC method,especially for stiff problems.Furthermore,we resolve the issue of low accuracy arising from the singularity of the solutions by using a geometric mesh,leading to highly accurate solutions compared to uniform mesh solutions at almost the same computational cost.Moreover,for long-time integration of FDEs,using the geometric mesh leads to great computational savings as the total number of degrees of freedom required is relatively small.
