Basing on the relationship between ill-conditioned linear systems of algebraic equations and stiff systems of ordinary differelltial equations and using an explicit single step method with 2-order for stiff systems, t...Basing on the relationship between ill-conditioned linear systems of algebraic equations and stiff systems of ordinary differelltial equations and using an explicit single step method with 2-order for stiff systems, the numerical solution of the initial value problems relative to the ill-conditioned linear systems of algebraic equations is obtained. So does the approkimation solution of the ill-conditioned linear systems of algebraic equations.展开更多
In this paper we modify the EBDF method using the NDFs as predictors instead of BDFs. This modification, that we call ENDF, implies the local truncation error being smaller than in the EBDF method without losing too m...In this paper we modify the EBDF method using the NDFs as predictors instead of BDFs. This modification, that we call ENDF, implies the local truncation error being smaller than in the EBDF method without losing too much stability. We will also introduce two more changes, called ENBDF and EBNDF methods. In the first one, the NDF method is used as the first predictor and the BDF as the second predictor. In the EBNDF, the BDF is the first predictor and the NDF is the second one. In both modifications the local truncation error is smaller than in the EBDF. Moreover, the EBNDF method has a larger stability region than the EBDF.展开更多
The improvements of high-throughput experimental devices such as microarray and mass spectrometry have allowed an effective acquisition of biological comprehensive data which include genome, transcriptome, proteome, a...The improvements of high-throughput experimental devices such as microarray and mass spectrometry have allowed an effective acquisition of biological comprehensive data which include genome, transcriptome, proteome, and metabolome (multi-layered omics data). In Systems Biology, we try to elucidate various dynamical characteristics of biological functions with applying the omics data to detailed mathematical model based on the central dogma. However, such mathematical models possess multi-time-scale properties which are often accompanied by time-scale differences seen among biological layers. The differences cause time stiff problem, and have a grave influence on numerical calculation stability. In the present conventional method, the time stiff problem remained because the calculation of all layers was implemented by adaptive time step sizes of the smallest time-scale layer to ensure stability and maintain calculation accuracy. In this paper, we designed and developed an effective numerical calculation method to improve the time stiff problem. This method consisted of ahead, backward, and cumulative algorithms. Both ahead and cumulative algorithms enhanced calculation efficiency of numerical calculations via adjustments of step sizes of each layer, and reduced the number of numerical calculations required for multi-time-scale models with the time stiff problem. Backward algorithm ensured calculation accuracy in the multi-time-scale models. In case studies which were focused on three layers system with 60 times difference in time-scale order in between layers, a proposed method had almost the same calculation accuracy compared with the conventional method in spite of a reduction of the total amount of the number of numerical calculations. Accordingly, the proposed method is useful in a numerical analysis of multi-time-scale models with time stiff problem.展开更多
Deals with the solution of partitioned systems of nonlinear stiff differential equations. Discussion on parallel compound methods; Nonstiff equations; Order conditions of the parallel compound methods (PCM); Numerical...Deals with the solution of partitioned systems of nonlinear stiff differential equations. Discussion on parallel compound methods; Nonstiff equations; Order conditions of the parallel compound methods (PCM); Numerical stability of PCM.展开更多
In this article we propose a higher-order space-time conservative method for hyperbolic systems with stiff and non stiff source terms as well as relaxation systems.We call the scheme a slope propagation(SP)method.It i...In this article we propose a higher-order space-time conservative method for hyperbolic systems with stiff and non stiff source terms as well as relaxation systems.We call the scheme a slope propagation(SP)method.It is an extension of our scheme derived for homogeneous hyperbolic systems[1].In the present inhomogeneous systems the relaxation time may vary from order of one to a very small value.These small values make the relaxation term stronger and highly stiff.In such situations underresolved numerical schemes may produce spurious numerical results.However,our present scheme has the capability to correctly capture the behavior of the physical phenomena with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved.The scheme treats the space and time in a unified manner.The flow variables and their slopes are the basic unknowns in the scheme.The source term is treated by its volumetric integration over the space-time control volume and is a direct part of the overall space-time flux balance.We use two approaches for the slope calculations of the flow variables,the first one results directly from the flux balance over the control volumes,while in the second one we use a finite difference approach.The main features of the scheme are its simplicity,its Jacobian-free and Riemann solver-free recipe,as well as its efficiency and high of order accuracy.In particular we show that the scheme has a discrete analog of the continuous asymptotic limit.We have implemented our scheme for various test models available in the literature such as the Broadwell model,the extended thermodynamics equations,the shallow water equations,traffic flow and the Euler equations with heat transfer.The numerical results validate the accuracy,versatility and robustness of the present scheme.展开更多
Stabilized or Chebyshev explicit methods have been widely used in the past to solve stiff ordinary differential equations. Making use of special properties of Chebyshev-like polynomials, these methods have favorable s...Stabilized or Chebyshev explicit methods have been widely used in the past to solve stiff ordinary differential equations. Making use of special properties of Chebyshev-like polynomials, these methods have favorable stability properties compared to standard explicit methods while remaining explicit. A new class of such methods, called ROCK, introduced in [Numer. Math., 90, 1-18, 2001] has recently been extended to stiff stochastic differential equations under the name S-ROCK [C. R. Acad. Sci. Paris, 345（10）, 2007 and Commun. Math. Sci, 6（4）, 2008]. In this paper we discuss the extension of the S-ROCK methods to systems with discrete noise and propose a new class of methods for such problems, the τ-ROCK methods. One motivation for such methods is the simulation of multi-scale or stiff chemical kinetic systems and such systems are the focus of this paper, but our new methods could potentially be interesting for other stiff systems with discrete noise. Two versions of the τ-ROCK methods are discussed and their stability behavior is analyzed on a test problem. Compared to the τ-leaping method, a significant speed-up can be achieved for some stiff kinetic systems. The behavior of the proposed methods are tested on several numerical experiments.展开更多
文摘Basing on the relationship between ill-conditioned linear systems of algebraic equations and stiff systems of ordinary differelltial equations and using an explicit single step method with 2-order for stiff systems, the numerical solution of the initial value problems relative to the ill-conditioned linear systems of algebraic equations is obtained. So does the approkimation solution of the ill-conditioned linear systems of algebraic equations.
文摘In this paper we modify the EBDF method using the NDFs as predictors instead of BDFs. This modification, that we call ENDF, implies the local truncation error being smaller than in the EBDF method without losing too much stability. We will also introduce two more changes, called ENBDF and EBNDF methods. In the first one, the NDF method is used as the first predictor and the BDF as the second predictor. In the EBNDF, the BDF is the first predictor and the NDF is the second one. In both modifications the local truncation error is smaller than in the EBDF. Moreover, the EBNDF method has a larger stability region than the EBDF.
文摘The improvements of high-throughput experimental devices such as microarray and mass spectrometry have allowed an effective acquisition of biological comprehensive data which include genome, transcriptome, proteome, and metabolome (multi-layered omics data). In Systems Biology, we try to elucidate various dynamical characteristics of biological functions with applying the omics data to detailed mathematical model based on the central dogma. However, such mathematical models possess multi-time-scale properties which are often accompanied by time-scale differences seen among biological layers. The differences cause time stiff problem, and have a grave influence on numerical calculation stability. In the present conventional method, the time stiff problem remained because the calculation of all layers was implemented by adaptive time step sizes of the smallest time-scale layer to ensure stability and maintain calculation accuracy. In this paper, we designed and developed an effective numerical calculation method to improve the time stiff problem. This method consisted of ahead, backward, and cumulative algorithms. Both ahead and cumulative algorithms enhanced calculation efficiency of numerical calculations via adjustments of step sizes of each layer, and reduced the number of numerical calculations required for multi-time-scale models with the time stiff problem. Backward algorithm ensured calculation accuracy in the multi-time-scale models. In case studies which were focused on three layers system with 60 times difference in time-scale order in between layers, a proposed method had almost the same calculation accuracy compared with the conventional method in spite of a reduction of the total amount of the number of numerical calculations. Accordingly, the proposed method is useful in a numerical analysis of multi-time-scale models with time stiff problem.
文摘Deals with the solution of partitioned systems of nonlinear stiff differential equations. Discussion on parallel compound methods; Nonstiff equations; Order conditions of the parallel compound methods (PCM); Numerical stability of PCM.
文摘In this article we propose a higher-order space-time conservative method for hyperbolic systems with stiff and non stiff source terms as well as relaxation systems.We call the scheme a slope propagation(SP)method.It is an extension of our scheme derived for homogeneous hyperbolic systems[1].In the present inhomogeneous systems the relaxation time may vary from order of one to a very small value.These small values make the relaxation term stronger and highly stiff.In such situations underresolved numerical schemes may produce spurious numerical results.However,our present scheme has the capability to correctly capture the behavior of the physical phenomena with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved.The scheme treats the space and time in a unified manner.The flow variables and their slopes are the basic unknowns in the scheme.The source term is treated by its volumetric integration over the space-time control volume and is a direct part of the overall space-time flux balance.We use two approaches for the slope calculations of the flow variables,the first one results directly from the flux balance over the control volumes,while in the second one we use a finite difference approach.The main features of the scheme are its simplicity,its Jacobian-free and Riemann solver-free recipe,as well as its efficiency and high of order accuracy.In particular we show that the scheme has a discrete analog of the continuous asymptotic limit.We have implemented our scheme for various test models available in the literature such as the Broadwell model,the extended thermodynamics equations,the shallow water equations,traffic flow and the Euler equations with heat transfer.The numerical results validate the accuracy,versatility and robustness of the present scheme.
基金supported by the National Science Foundation of China under grant 10871010the National Basic Research Program under grant 2005CB321704
文摘Stabilized or Chebyshev explicit methods have been widely used in the past to solve stiff ordinary differential equations. Making use of special properties of Chebyshev-like polynomials, these methods have favorable stability properties compared to standard explicit methods while remaining explicit. A new class of such methods, called ROCK, introduced in [Numer. Math., 90, 1-18, 2001] has recently been extended to stiff stochastic differential equations under the name S-ROCK [C. R. Acad. Sci. Paris, 345（10）, 2007 and Commun. Math. Sci, 6（4）, 2008]. In this paper we discuss the extension of the S-ROCK methods to systems with discrete noise and propose a new class of methods for such problems, the τ-ROCK methods. One motivation for such methods is the simulation of multi-scale or stiff chemical kinetic systems and such systems are the focus of this paper, but our new methods could potentially be interesting for other stiff systems with discrete noise. Two versions of the τ-ROCK methods are discussed and their stability behavior is analyzed on a test problem. Compared to the τ-leaping method, a significant speed-up can be achieved for some stiff kinetic systems. The behavior of the proposed methods are tested on several numerical experiments.
