查看更多>>摘要：We consider the application of s-stage implicit Runge-Kutta methods to ordinary differential equations (ODEs). We consider starting approximations based on values from the previous step to obtain an accurate initial guess for the internal stages of the current step. To simplify the analysis of those starting approximations we compare the expansions of the starting approximation and of the exact value of the internal stages at the initial value x_n of the current step and not at the initial value x_(n-1) of the previous step. In particular, for the starting approximation we make use of the expansion of the reverse IRK method from the initial value x_n of the current step with a negative step size. This simplifies considerably the expression of the order conditions. As a consequence it allows us to give more general and precise statements about the existence and uniqueness of a starting approximation of a given order for IRK methods satisfying the simplifying assumptions B(p) and C(q). In particular we show under certain assumptions the nonexistence of starting approximations of order s+ 1 for the type of starting approximations considered.
查看更多>>摘要：We consider the application of s-stage implicit Runge-Kutta methods to ordinary differential equations (ODEs). We consider starting approximations based on values from the previous step to obtain an accurate initial guess for the internal stages of the current step. To simplify the analysis of those starting approximations we compare the expansions of the starting approximation and of the exact value of the internal stages at the initial value x_n of the current step and not at the initial value x_(n-1) of the previous step. In particular, for the starting approximation we make use of the expansion of the reverse IRK method from the initial value x_n of the current step with a negative step size. This simplifies considerably the expression of the order conditions. As a consequence it allows us to give more general and precise statements about the existence and uniqueness of a starting approximation of a given order for IRK methods satisfying the simplifying assumptions B(p) and C(q). In particular we show under certain assumptions the nonexistence of starting approximations of order s+ 1 for the type of starting approximations considered.
查看更多>>摘要：Autoresonance is a phenomenon of physical interest that may take place when a nonlinear oscillator is forced at a frequency that varies slowly. The stroboscopic averaging method (SAM), which provides an efficient numerical technique for the integration of highly oscillatory systems, cannot be used directly to study autoresonance due to the slow changes of the forcing frequency. We study how to modify SAM to cater for such slow variations. Numerical experiments show the computational advantages of using SAM.
查看更多>>摘要：Autoresonance is a phenomenon of physical interest that may take place when a nonlinear oscillator is forced at a frequency that varies slowly. The stroboscopic averaging method (SAM), which provides an efficient numerical technique for the integration of highly oscillatory systems, cannot be used directly to study autoresonance due to the slow changes of the forcing frequency. We study how to modify SAM to cater for such slow variations. Numerical experiments show the computational advantages of using SAM.
查看更多>>摘要：In this paper, we consider a system of accelerated and general fully non-linear discrete equations depending on a parameter σ lying inside an interval [σ~-,σ~+]. For σ ∈ (σ~-,σ~+), our non-linearity is bistable and for σ =σ~±, it is monostable. Two results are obtained: the first one is to derive properties of the velocity function associated to the existence of traveling waves in the bistable regimes. The second one is to construct traveling waves in the monostable regimes. Our approach is to consider the monostable regimes as the limit of bistable ones. As far as we know, this is the first result concerning traveling waves for accelerated, general and monostable fully-nonlinear discrete system.
查看更多>>摘要：In this paper, we consider a system of accelerated and general fully non-linear discrete equations depending on a parameter σ lying inside an interval [σ~-,σ~+]. For σ ∈ (σ~-,σ~+), our non-linearity is bistable and for σ =σ~±, it is monostable. Two results are obtained: the first one is to derive properties of the velocity function associated to the existence of traveling waves in the bistable regimes. The second one is to construct traveling waves in the monostable regimes. Our approach is to consider the monostable regimes as the limit of bistable ones. As far as we know, this is the first result concerning traveling waves for accelerated, general and monostable fully-nonlinear discrete system.
查看更多>>摘要：In the study of implicit iterations for the two dimensional heat and Helmhotz equations, one constructs a splitting of the form A = U + V where U and V are the difference approximations parallel to the x and y axis, respectively. In the commutative case for U and V, several investigations have taken place. For the noncommutative case, a symmetric positive definite matrix F is found, such that UFV = VFU and then, the investigations use this to address the non-commutative nature of U with V. The purpose of this paper is to study the same problem type for the commutativity of H and S in the HSS splitting of A = H + S, where H and S are the symmetric and skew-symmetric parts of A, respectively. Although throughout the literature in C~(n×n), the H of HSS stands for hermitian, we use it in the symmetric matrix case. One then applies this result to the HSS iteration method. Since it is well known that the commutativity of U and V plays an important role in the analysis of ADI methods; especially for the solution of the Helmhotz Equation, one hopes that this commutativity will improve performance of the HSS method. This extension is similar to that of Wachspress and Habetler's variation of the Peaceman-Rachford method. Along the way, suitable conditions are found for A, which yield a symmetric non-zero matrix P = F~(1/2) for which N = PAP is a normal matrix.
查看更多>>摘要：In the study of implicit iterations for the two dimensional heat and Helmhotz equations, one constructs a splitting of the form A = U + V where U and V are the difference approximations parallel to the x and y axis, respectively. In the commutative case for U and V, several investigations have taken place. For the noncommutative case, a symmetric positive definite matrix F is found, such that UFV = VFU and then, the investigations use this to address the non-commutative nature of U with V. The purpose of this paper is to study the same problem type for the commutativity of H and S in the HSS splitting of A = H + S, where H and S are the symmetric and skew-symmetric parts of A, respectively. Although throughout the literature in C~(n×n), the H of HSS stands for hermitian, we use it in the symmetric matrix case. One then applies this result to the HSS iteration method. Since it is well known that the commutativity of U and V plays an important role in the analysis of ADI methods; especially for the solution of the Helmhotz Equation, one hopes that this commutativity will improve performance of the HSS method. This extension is similar to that of Wachspress and Habetler's variation of the Peaceman-Rachford method. Along the way, suitable conditions are found for A, which yield a symmetric non-zero matrix P = F~(1/2) for which N = PAP is a normal matrix.
查看更多>>摘要：In this paper, a relaxation-type high-order compact finite difference (RCFD) scheme is proposed for the one-dimensional nonlinear Schrodinger equation. More specifically, the relaxation approach combined with the Crank-Nicolson formula is utilized for time discretization, and fourth-order compact difference method is applied for space discretization. The scheme is linear, decoupled, and can be solved sequentially with respect to the primal and relaxation variables, which avoids solving large-scale nonlinear algebraic systems resulting in fully implicit numerical schemes. Furthermore, the developed scheme is shown to preserve both mass and energy at the discrete level. Most importantly, with the help of a discrete elliptic projection and a cutoff numerical technique, the existence and uniqueness of the high-order RCFD scheme are ensured, and unconditional optimal-order error estimate in discrete maximum-norm is rigorously established. Finally, several numerical experiments are given to support the theoretical findings, and comparisons with other methods are also presented to show the efficiency and effectiveness of our method.
查看更多>>摘要：In this paper, a relaxation-type high-order compact finite difference (RCFD) scheme is proposed for the one-dimensional nonlinear Schrodinger equation. More specifically, the relaxation approach combined with the Crank-Nicolson formula is utilized for time discretization, and fourth-order compact difference method is applied for space discretization. The scheme is linear, decoupled, and can be solved sequentially with respect to the primal and relaxation variables, which avoids solving large-scale nonlinear algebraic systems resulting in fully implicit numerical schemes. Furthermore, the developed scheme is shown to preserve both mass and energy at the discrete level. Most importantly, with the help of a discrete elliptic projection and a cutoff numerical technique, the existence and uniqueness of the high-order RCFD scheme are ensured, and unconditional optimal-order error estimate in discrete maximum-norm is rigorously established. Finally, several numerical experiments are given to support the theoretical findings, and comparisons with other methods are also presented to show the efficiency and effectiveness of our method.
NETL
NSTL
Elsevier