Numerical Solution Of G-heat Equation Based On Finite Difference Method

In this paper,we mainly explore the finite difference method to solve the problem of one-dimensional G-heat equation.We propose several numerical difference formats and then analyze the accuracy of each format by numerical examples.Noting that the G-heat equation is a parabolic equation with variable coef-ficients,this paper first reviews the common finite difference formats for solving the constant coefficient parabolic equation and the stability conditions for each format.These include the forward difference format,the backward difference format,the six-point symmetric format(Crank-Nicolson format),the three-level explicit format(Richardson format),and the three-level implicit format,and then we explore how these formats can be applied to solve the G-heat equation.When the finite difference format of the parabolic partial differential equa-tion is applied to solve the G-heat equation,the forward difference format and the three-level explicit format can be used directly for the solution.When the re-maining three implicit formats are used to solve the G-heat equation numerically,the coefficients of the G-equation are not directly available due to the positive and negative second-order derivative terms themselves,which makes the calcula-tion very difficult.Therefore,in this paper,we innovatively improve the iteration coefficients in the three implicit formats and propose a semi-implicit format that can be used to solve the G-equation:backward difference format(semi-implicit),six-point difference format(semi-implicit)and three-level implicit format(semi-implicit).Based on the Fourier method and von Neumann's conditional stability crite-rion,the stability conditions for each format are derived in this paper:1.The forward differential format is stable for the net ratio rmax=?2?/2h2<1/2:2.Backward differential format(semi-implicit),six-point differential format(semi-implicit)and three-layer implicit format(semi-implicit)are stable for any net ratio;3.The three-layer explicit format(Richardson format)is unstable for any net ratio.Finally,in this paper,we solve the G-heat equation for different initial values using the four stable formats mentioned above.We consider the G-heat equation when the initial value is a continuous function ?(x)=(x+a)3,and the initial value is a discrete schematic function ?(x)=I{x>a},?(x)=I{x<?},?(x)=I{|x|>a},respectively,and analyzing the error data of the numerical solutions we obtain the following conclusions:1.The forwaxd difference format has the same order of solution error as the trinomial tree method used by Hu M;2.The accuracy of the three-level implicit format(semi-implicit)and the six-point difference format(semi-implicit)is higher than that of the backward difference format(semi-implicit)and the forward difference format,and the accuracy of the numerical solution in each format improves significantly with increasing time step.3.When the four formats are used to solve the G-heat equation with the initial value of ?(x)=I{|x|>a},within the error range,the numerical solutions all satisfy the asymptotic inequality of the true solution.Therefore,we suggest that the three-level implicit format(semi-implicit)and the six-point difference format(semi-implicit)should be preferred when solving the G-heat equation with unknown true solutions.