We introduce a new Euler-type scheme and its iterative algorithm for solving weakly coupled forward-backward stochastic differential equations （FBSDEs）. Although the schemes share some common features with the ones ...We introduce a new Euler-type scheme and its iterative algorithm for solving weakly coupled forward-backward stochastic differential equations （FBSDEs）. Although the schemes share some common features with the ones proposed by C. Bender and J. Zhang [Ann. Appl. Probab., 2008, 18： 143-177], less computational work is needed for our method. For both our schemes and the ones proposed by Bender and Zhang, we rigorously obtain first-order error estimates, which improve the half-order error estimates of Bender and Zhang. Moreover, numerical tests are given to demonstrate the first-order accuracy of the schemes.展开更多
The time discretization in the Discontinuous Galerkin （DG） scheme has been traditionally based on the Total Variation Diminishing （TVD） second-order Runge-Kutta （RK2） scheme. Computational efficiency and accurac...The time discretization in the Discontinuous Galerkin （DG） scheme has been traditionally based on the Total Variation Diminishing （TVD） second-order Runge-Kutta （RK2） scheme. Computational efficiency and accuracy with the Euler Forward （EF） and the TVD second-order RK2 time stepping schemes in the DG method are investigated in this work. Numerical tests are condu- cted with the scalar Burgers equation, 1-D and 2-D shallow water flow equations. The maximum Courant number or time step size required for stability for the EF scheme and RK2 scheme with different slope limiters are compared. Numerical results show that the slope limiters affect the stability requirement in the DG method. The RK2 scheme is generally more diffusive than the EF scheme, and the RK2 scheme allows larger time step sizes. The EF scheme is found to be more efficient and accurate than the RK2 scheme in the DG method in computation.展开更多
基金Acknowledgements The authors would like to thank the referees for the valuable comments, which improved the paper a lot. This work was partially supported by the National Natural Science Foundations of China （Grant Nos. 91130003, 11171189） and the Natural Science Foundation of Shandong Province （No. ZR2011AZ002）.
文摘We introduce a new Euler-type scheme and its iterative algorithm for solving weakly coupled forward-backward stochastic differential equations （FBSDEs）. Although the schemes share some common features with the ones proposed by C. Bender and J. Zhang [Ann. Appl. Probab., 2008, 18： 143-177], less computational work is needed for our method. For both our schemes and the ones proposed by Bender and Zhang, we rigorously obtain first-order error estimates, which improve the half-order error estimates of Bender and Zhang. Moreover, numerical tests are given to demonstrate the first-order accuracy of the schemes.
文摘The time discretization in the Discontinuous Galerkin （DG） scheme has been traditionally based on the Total Variation Diminishing （TVD） second-order Runge-Kutta （RK2） scheme. Computational efficiency and accuracy with the Euler Forward （EF） and the TVD second-order RK2 time stepping schemes in the DG method are investigated in this work. Numerical tests are condu- cted with the scalar Burgers equation, 1-D and 2-D shallow water flow equations. The maximum Courant number or time step size required for stability for the EF scheme and RK2 scheme with different slope limiters are compared. Numerical results show that the slope limiters affect the stability requirement in the DG method. The RK2 scheme is generally more diffusive than the EF scheme, and the RK2 scheme allows larger time step sizes. The EF scheme is found to be more efficient and accurate than the RK2 scheme in the DG method in computation.
