As is known,there exist numerous alternating direction implicit(ADI)schemes for the two-dimensional linear time fractional partial differential equations(PDEs).However,if the ADI schemes for linear problems combined w...As is known,there exist numerous alternating direction implicit(ADI)schemes for the two-dimensional linear time fractional partial differential equations(PDEs).However,if the ADI schemes for linear problems combined with local linearization techniques are applied to solve nonlinear problems,the stability and convergence of the methods are often not clear.In this paper,two ADI schemes are developed for solving the two-dimensional time fractional nonlinear super-diffusion equations based on their equivalent partial integrodifferential equations.In these two schemes,the standard second-order central difference approximation is used for the spatial discretization,and the classical first-order approximation is applied to discretize the Riemann-Liouville fractional integral in time.The solvability,unconditional stability and L2 norm convergence of the proposed ADI schemes are proved rigorously.The convergence order of the schemes is 0(τ+hx^2+hy^2),where τ is the temporal mesh size,hx and hy are spatial mesh sizes in the x and y directions,respectively.Finally,numerical experiments are carried out to support the theoretical results and demonstrate the performances of two ADI schemes.展开更多
We design a nunchakus-like tracer and investigate its self-adaptive behavior in an active Brownian particle(ABP)bath via systematically tuning the self-propelled capability and density of ABPs.Specifically,the nunchak...We design a nunchakus-like tracer and investigate its self-adaptive behavior in an active Brownian particle(ABP)bath via systematically tuning the self-propelled capability and density of ABPs.Specifically,the nunchakus-like tracer will have a stable wedge-like shape in the ABP bath when the self-propelled force is high enough.We analyze the angle between the two arms of the tracer and the velocity of the joint point of the tracer.The angle exhibits a non-monotonic phenomenon as a function of active force.However,it increases with density of ABPs increasing monotonically.A simple linear relationship between the velocity and the self-propelled force is found under the highly active force.In other words,the joint points of the tracer diffuse and the super-diffusive behavior can make the relation between the self-propelled force and the density of ABPs persist longer.In addition,we find that the tracer can flip at high density of ABPs.Our results also suggest the new self-adaptive model research of the transport properties in a non-equilibrium medium.展开更多
基金National Natural Science Foundation of China(Grant Nos.11701502 and 11771438).
文摘As is known,there exist numerous alternating direction implicit(ADI)schemes for the two-dimensional linear time fractional partial differential equations(PDEs).However,if the ADI schemes for linear problems combined with local linearization techniques are applied to solve nonlinear problems,the stability and convergence of the methods are often not clear.In this paper,two ADI schemes are developed for solving the two-dimensional time fractional nonlinear super-diffusion equations based on their equivalent partial integrodifferential equations.In these two schemes,the standard second-order central difference approximation is used for the spatial discretization,and the classical first-order approximation is applied to discretize the Riemann-Liouville fractional integral in time.The solvability,unconditional stability and L2 norm convergence of the proposed ADI schemes are proved rigorously.The convergence order of the schemes is 0(τ+hx^2+hy^2),where τ is the temporal mesh size,hx and hy are spatial mesh sizes in the x and y directions,respectively.Finally,numerical experiments are carried out to support the theoretical results and demonstrate the performances of two ADI schemes.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11804085 and 21674078)the Natural Science Foundation of the Jiangsu Provincial Higher Education Institutions,China(Grant No.21KJB140023)the Foundation of Jiangsu Provincial Innovation and Entrepreneurship Doctor,China(Grant No.JSSCBS20211147)。
文摘We design a nunchakus-like tracer and investigate its self-adaptive behavior in an active Brownian particle(ABP)bath via systematically tuning the self-propelled capability and density of ABPs.Specifically,the nunchakus-like tracer will have a stable wedge-like shape in the ABP bath when the self-propelled force is high enough.We analyze the angle between the two arms of the tracer and the velocity of the joint point of the tracer.The angle exhibits a non-monotonic phenomenon as a function of active force.However,it increases with density of ABPs increasing monotonically.A simple linear relationship between the velocity and the self-propelled force is found under the highly active force.In other words,the joint points of the tracer diffuse and the super-diffusive behavior can make the relation between the self-propelled force and the density of ABPs persist longer.In addition,we find that the tracer can flip at high density of ABPs.Our results also suggest the new self-adaptive model research of the transport properties in a non-equilibrium medium.
