In this paper, based on the implicit Runge-Kutta(IRK) methods, we derive a class of parallel scheme that can be implemented on the parallel computers with Ns(N is a positive even number) processors efficiently, and di...In this paper, based on the implicit Runge-Kutta(IRK) methods, we derive a class of parallel scheme that can be implemented on the parallel computers with Ns(N is a positive even number) processors efficiently, and discuss the iteratively B-convergence of the Newton iterative process for solving the algebraic equations of the scheme, secondly we present a strategy providing initial values parallelly for the iterative process. Finally, some numerical results show that our parallel scheme is higher efficient as N is not so large.展开更多
In this paper, the Ito-Taylor expansion of stochastic differential equation is briefly introduced. The colored rooted tree theory is applied to derive strong order 1.0 implicit stochastic Runge-Kutta method(SRK). Two ...In this paper, the Ito-Taylor expansion of stochastic differential equation is briefly introduced. The colored rooted tree theory is applied to derive strong order 1.0 implicit stochastic Runge-Kutta method(SRK). Two fully implicit schemes are presented and their stability qualities are discussed. And the numerical report illustrates the better numerical behavior.展开更多
This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either...This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.展开更多
Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were previously proposed and analyzed.These specially designed methods use reduced precision for the implic...Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were previously proposed and analyzed.These specially designed methods use reduced precision for the implicit computations and full precision for the explicit computations.In this work,we analyze the stability properties of these methods and their sensitivity to the low-precision rounding errors,and demonstrate their performance in terms of accuracy and efficiency.We develop codes in FORTRAN and Julia to solve nonlinear systems of ODEs and PDEs using the mixed-precision additive Runge-Kutta(MP-ARK)methods.The convergence,accuracy,and runtime of these methods are explored.We show that for a given level of accuracy,suitably chosen MP-ARK methods may provide significant reductions in runtime.展开更多
It is well known that mono-implicit Runge-Kutta methods have been applied in the efficient numerical solution of initial or boundary value problems of ordinary differential equations. Burrage (1994) has shown that the...It is well known that mono-implicit Runge-Kutta methods have been applied in the efficient numerical solution of initial or boundary value problems of ordinary differential equations. Burrage (1994) has shown that the order of an s-stage monoimplicit Runge-Kutta method is at most s+1 and the stage order is at most 3. In this paper, it is shown that the order of an s-stage mono-implicit Runge-Kutta method being algebraically stable is at most min((s) over tilde, 4), and the stage order together with the optimal B-convergence order is at most min(s, 2), where [GRAPHICS]展开更多
This article deals with a class of numerical methods for retarded differential algebraic systems with time-variable delay. The methods can be viewed as a combination of Runge-Kutta methods and Lagrange interpolation. ...This article deals with a class of numerical methods for retarded differential algebraic systems with time-variable delay. The methods can be viewed as a combination of Runge-Kutta methods and Lagrange interpolation. A new convergence concept, called DA-convergence, is introduced. The DA-convergence result for the methods is derived. At the end, a numerical example is given to verify the computational effectiveness and the theoretical result.展开更多
Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic ...Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations (PDEs) derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.展开更多
This paper discusses a kind of implicit iterative methods with some variable parameters, which are called control parameters, for solving ill-posed operator equations. The theoretical results show that the new methods...This paper discusses a kind of implicit iterative methods with some variable parameters, which are called control parameters, for solving ill-posed operator equations. The theoretical results show that the new methods always lead to optimal convergence rates and have some other important features, especially the methods can be implemented parallelly.展开更多
The compact implicit integration factor (cIIF) method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF me...The compact implicit integration factor (cIIF) method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger (NLS) equation and the complex Ginzburg-Landau (GL) equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.展开更多
Projected Runge-Kutta (R-K) methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations (DAEs) in the Heisenberg form, are establi...Projected Runge-Kutta (R-K) methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations (DAEs) in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.展开更多
An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods (TSCRK) is presented. Their construction, their order conditio...An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods (TSCRK) is presented. Their construction, their order conditions and their convergence are studied. The two-step continuity Runge-Kutta methods possess good numerical stability properties and higher stage-order, and keep the explicit process of computing the Runge-Kutta stages. The numerical experiments show that the TSCRK methods are efficient.展开更多
The aim of this paper is to study the asymptotic stability properties of Runge Kutta(R-K) methods for neutral differential equations(NDDEs) when they are applied to the linear test equation of the form: y′(t)=ay(t)...The aim of this paper is to study the asymptotic stability properties of Runge Kutta(R-K) methods for neutral differential equations(NDDEs) when they are applied to the linear test equation of the form: y′(t)=ay(t)+by(t-τ)+cy’(t-τ), t>0, y(t)=g(t), -τ≤t≤0, with a,b,c∈[FK(W+3mm\.3mm][TPP129A,+3mm?3mm,BP], τ>0 and g(t) is a continuous real value function. In this paper we are concerned with the dependence of stability region on a fixed but arbitrary delay τ. In fact, it is one of the N.Guglielmi open problems to investigate the delay dependent stability analysis for NDDEs. The results that the 2,3 stages non natural R-K methods are unstable as Radau IA and Lobatto IIIC are proved. And the s stages Radau IIA methods are unstable, however all Gauss methods are compatible.展开更多
In this paper the existence and uniqueness of the solution of implicit hybrid methods(IHMs)for solving the initial value problems(IVPs)of stiff ordinary differential equations(ODEs)is considered.We provide the coeffic...In this paper the existence and uniqueness of the solution of implicit hybrid methods(IHMs)for solving the initial value problems(IVPs)of stiff ordinary differential equations(ODEs)is considered.We provide the coefficient condition and its judging criterion as well as the righthand condition to ensure the existing solution uniquely.展开更多
In this paper two implicit 2-step hybrid methods are proposed! one has order five, the other six. The stability properties of the methods are analysed. The 5th order method is proved to be A-stable and the 6th order o...In this paper two implicit 2-step hybrid methods are proposed! one has order five, the other six. The stability properties of the methods are analysed. The 5th order method is proved to be A-stable and the 6th order one is not, but still has a relatively large region of absolute stability. The implementation of the 5th order method is also discussed.展开更多
For differential equations with piecewise constant arguments of advanced type, numerical stability and oscillations of Runge-Kutta methods are investigated. The necessary and sufficient conditions under which the nume...For differential equations with piecewise constant arguments of advanced type, numerical stability and oscillations of Runge-Kutta methods are investigated. The necessary and sufficient conditions under which the numerical stability region contains the analytic stability region are given. The conditions of oscillations for the Runge-Kutta methods are obtained also. We prove that the Runge-Kutta methods preserve the oscillations of the analytic solution. Moreover, the relationship between stability and oscillations is discussed. Several numerical examples which confirm the results of our analysis are presented.展开更多
In this paper we discuss diagonally implicit and semi-implicit methods based on the three-stage stiffly accurate Runge-Kutta methods for solving Stratonovich stochastic differential equations(SDEs).Two methods,a thr...In this paper we discuss diagonally implicit and semi-implicit methods based on the three-stage stiffly accurate Runge-Kutta methods for solving Stratonovich stochastic differential equations(SDEs).Two methods,a three-stage stiffly accurate semi-implicit(SASI3) method and a three-stage stiffly accurate diagonally implicit (SADI3) method,are constructed in this paper.In particular,the truncated random variable is used in the implicit method.The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of stiff SDEs.展开更多
A series of sufficient and necessary conditions for the algebraic stability of multistepRunge-Kutta methods is obtained, most of which can be regarded as extension of the relevant results available for Runge-Kutta met...A series of sufficient and necessary conditions for the algebraic stability of multistepRunge-Kutta methods is obtained, most of which can be regarded as extension of the relevant results available for Runge-Kutta methods, especially, for Radau Ⅰ A, Radau Ⅱ A and Gaussian Runge-Kutta methods.展开更多
In this paper, a rather general class of explicit parallel multistep Runge-Kutta methods is constructed for solving initial value problem of ordinary differential equations. Also, the corresponding convergence and sta...In this paper, a rather general class of explicit parallel multistep Runge-Kutta methods is constructed for solving initial value problem of ordinary differential equations. Also, the corresponding convergence and stability are analysed. Several parallel computational formulae are given. The numerical experiments, including accuracy, speedup, and efficiency tests show that the methods are efficient.展开更多
A time discretization method is called strongly stable(or monotone),if the norm of its numerical solution is nonincreasing.Although this property is desirable in various of contexts,many explicit Runge-Kutta(RK)method...A time discretization method is called strongly stable(or monotone),if the norm of its numerical solution is nonincreasing.Although this property is desirable in various of contexts,many explicit Runge-Kutta(RK)methods may fail to preserve it.In this paper,we enforce strong stability by modifying the method with superviscosity,which is a numerical technique commonly used in spectral methods.Our main focus is on strong stability under the inner-product norm for linear problems with possibly non-normal operators.We propose two approaches for stabilization:the modified method and the filtering method.The modified method is achieved by modifying the semi-negative operator with a high order superviscosity term;the filtering method is to post-process the solution by solving a diffusive or dispersive problem with small superviscosity.For linear problems,most explicit RK methods can be stabilized with either approach without accuracy degeneration.Furthermore,we prove a sharp bound(up to an equal sign)on diffusive superviscosity for ensuring strong stability.For nonlinear problems,a filtering method is investigated.Numerical examples with linear non-normal ordinary differential equation systems and for discontinuous Galerkin approximations of conservation laws are performed to validate our analysis and to test the performance.展开更多
Numerical dispersion relation of the multi-symplectic Runge-Kutta (MSRK) method for linear Hamiltonian PDEs is derived in the present paper, which is shown to be a discrete counterpart to that possessed by the diffe...Numerical dispersion relation of the multi-symplectic Runge-Kutta (MSRK) method for linear Hamiltonian PDEs is derived in the present paper, which is shown to be a discrete counterpart to that possessed by the differential equation. This provides further understanding of MSRK methods. However, much still remains to be investigated further.展开更多
基金national natural science foundation natural science foundation of Gansu province.
文摘In this paper, based on the implicit Runge-Kutta(IRK) methods, we derive a class of parallel scheme that can be implemented on the parallel computers with Ns(N is a positive even number) processors efficiently, and discuss the iteratively B-convergence of the Newton iterative process for solving the algebraic equations of the scheme, secondly we present a strategy providing initial values parallelly for the iterative process. Finally, some numerical results show that our parallel scheme is higher efficient as N is not so large.
文摘In this paper, the Ito-Taylor expansion of stochastic differential equation is briefly introduced. The colored rooted tree theory is applied to derive strong order 1.0 implicit stochastic Runge-Kutta method(SRK). Two fully implicit schemes are presented and their stability qualities are discussed. And the numerical report illustrates the better numerical behavior.
基金supported by the NSF under Grant DMS-2208391sponsored by the NSF under Grant DMS-1753581.
文摘This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.
基金supported by ONR UMass Dartmouth Marine and UnderSea Technology(MUST)grant N00014-20-1-2849 under the project S31320000049160by DOE grant DE-SC0023164 sub-award RC114586-UMD+2 种基金by AFOSR grants FA9550-18-1-0383 and FA9550-23-1-0037supported by Michigan State University,by AFOSR grants FA9550-19-1-0281 and FA9550-18-1-0383by DOE grant DE-SC0023164.
文摘Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were previously proposed and analyzed.These specially designed methods use reduced precision for the implicit computations and full precision for the explicit computations.In this work,we analyze the stability properties of these methods and their sensitivity to the low-precision rounding errors,and demonstrate their performance in terms of accuracy and efficiency.We develop codes in FORTRAN and Julia to solve nonlinear systems of ODEs and PDEs using the mixed-precision additive Runge-Kutta(MP-ARK)methods.The convergence,accuracy,and runtime of these methods are explored.We show that for a given level of accuracy,suitably chosen MP-ARK methods may provide significant reductions in runtime.
文摘It is well known that mono-implicit Runge-Kutta methods have been applied in the efficient numerical solution of initial or boundary value problems of ordinary differential equations. Burrage (1994) has shown that the order of an s-stage monoimplicit Runge-Kutta method is at most s+1 and the stage order is at most 3. In this paper, it is shown that the order of an s-stage mono-implicit Runge-Kutta method being algebraically stable is at most min((s) over tilde, 4), and the stage order together with the optimal B-convergence order is at most min(s, 2), where [GRAPHICS]
文摘This article deals with a class of numerical methods for retarded differential algebraic systems with time-variable delay. The methods can be viewed as a combination of Runge-Kutta methods and Lagrange interpolation. A new convergence concept, called DA-convergence, is introduced. The DA-convergence result for the methods is derived. At the end, a numerical example is given to verify the computational effectiveness and the theoretical result.
基金supported by the National Natural Science Foundation of China (Nos. 10772147 and10632030)the Ph. D. Program Foundation of Ministry of Education of China (No. 20070699028)+2 种基金the Natural Science Foundation of Shaanxi Province of China (No. 2006A07)the Open Foundationof State Key Laboratory of Structural Analysis of Industrial Equipment (No. GZ0802)the Foundation for Fundamental Research of Northwestern Polytechnical University
文摘Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations (PDEs) derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.
基金This work was supported by the National Natural Science Foundation of China
文摘This paper discusses a kind of implicit iterative methods with some variable parameters, which are called control parameters, for solving ill-posed operator equations. The theoretical results show that the new methods always lead to optimal convergence rates and have some other important features, especially the methods can be implemented parallelly.
文摘The compact implicit integration factor (cIIF) method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger (NLS) equation and the complex Ginzburg-Landau (GL) equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.
基金Project supported by the National Natural Science Foundation of China(No.11432010)the Doctoral Program Foundation of Education Ministry of China(No.20126102110023)+2 种基金the 111Project of China(No.B07050)the Fundamental Research Funds for the Central Universities(No.310201401JCQ01001)the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University(No.CX201517)
文摘Projected Runge-Kutta (R-K) methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations (DAEs) in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.
文摘An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods (TSCRK) is presented. Their construction, their order conditions and their convergence are studied. The two-step continuity Runge-Kutta methods possess good numerical stability properties and higher stage-order, and keep the explicit process of computing the Runge-Kutta stages. The numerical experiments show that the TSCRK methods are efficient.
文摘The aim of this paper is to study the asymptotic stability properties of Runge Kutta(R-K) methods for neutral differential equations(NDDEs) when they are applied to the linear test equation of the form: y′(t)=ay(t)+by(t-τ)+cy’(t-τ), t>0, y(t)=g(t), -τ≤t≤0, with a,b,c∈[FK(W+3mm\.3mm][TPP129A,+3mm?3mm,BP], τ>0 and g(t) is a continuous real value function. In this paper we are concerned with the dependence of stability region on a fixed but arbitrary delay τ. In fact, it is one of the N.Guglielmi open problems to investigate the delay dependent stability analysis for NDDEs. The results that the 2,3 stages non natural R-K methods are unstable as Radau IA and Lobatto IIIC are proved. And the s stages Radau IIA methods are unstable, however all Gauss methods are compatible.
基金Supported by the national natural science foundation.
文摘In this paper the existence and uniqueness of the solution of implicit hybrid methods(IHMs)for solving the initial value problems(IVPs)of stiff ordinary differential equations(ODEs)is considered.We provide the coefficient condition and its judging criterion as well as the righthand condition to ensure the existing solution uniquely.
文摘In this paper two implicit 2-step hybrid methods are proposed! one has order five, the other six. The stability properties of the methods are analysed. The 5th order method is proved to be A-stable and the 6th order one is not, but still has a relatively large region of absolute stability. The implementation of the 5th order method is also discussed.
文摘For differential equations with piecewise constant arguments of advanced type, numerical stability and oscillations of Runge-Kutta methods are investigated. The necessary and sufficient conditions under which the numerical stability region contains the analytic stability region are given. The conditions of oscillations for the Runge-Kutta methods are obtained also. We prove that the Runge-Kutta methods preserve the oscillations of the analytic solution. Moreover, the relationship between stability and oscillations is discussed. Several numerical examples which confirm the results of our analysis are presented.
基金supported by the NSF(10926158) of ChinaDoctoral Fund(20090061120038) of Ministry of Education of ChinaBasic Scientific Research Foundation(200903287) of Jilin University
文摘In this paper we discuss diagonally implicit and semi-implicit methods based on the three-stage stiffly accurate Runge-Kutta methods for solving Stratonovich stochastic differential equations(SDEs).Two methods,a three-stage stiffly accurate semi-implicit(SASI3) method and a three-stage stiffly accurate diagonally implicit (SADI3) method,are constructed in this paper.In particular,the truncated random variable is used in the implicit method.The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of stiff SDEs.
文摘A series of sufficient and necessary conditions for the algebraic stability of multistepRunge-Kutta methods is obtained, most of which can be regarded as extension of the relevant results available for Runge-Kutta methods, especially, for Radau Ⅰ A, Radau Ⅱ A and Gaussian Runge-Kutta methods.
文摘In this paper, a rather general class of explicit parallel multistep Runge-Kutta methods is constructed for solving initial value problem of ordinary differential equations. Also, the corresponding convergence and stability are analysed. Several parallel computational formulae are given. The numerical experiments, including accuracy, speedup, and efficiency tests show that the methods are efficient.
基金supported by NSF Grants DMS-1719410 and DMS-2010107by AFOSR Grant FA9550-20-1-0055.
文摘A time discretization method is called strongly stable(or monotone),if the norm of its numerical solution is nonincreasing.Although this property is desirable in various of contexts,many explicit Runge-Kutta(RK)methods may fail to preserve it.In this paper,we enforce strong stability by modifying the method with superviscosity,which is a numerical technique commonly used in spectral methods.Our main focus is on strong stability under the inner-product norm for linear problems with possibly non-normal operators.We propose two approaches for stabilization:the modified method and the filtering method.The modified method is achieved by modifying the semi-negative operator with a high order superviscosity term;the filtering method is to post-process the solution by solving a diffusive or dispersive problem with small superviscosity.For linear problems,most explicit RK methods can be stabilized with either approach without accuracy degeneration.Furthermore,we prove a sharp bound(up to an equal sign)on diffusive superviscosity for ensuring strong stability.For nonlinear problems,a filtering method is investigated.Numerical examples with linear non-normal ordinary differential equation systems and for discontinuous Galerkin approximations of conservation laws are performed to validate our analysis and to test the performance.
基金The NNSF (10471054) of Chinathe China Postdoctoral Science Foundationthe Youth Foundation of Institute of Mathematics, Jilin University
文摘Numerical dispersion relation of the multi-symplectic Runge-Kutta (MSRK) method for linear Hamiltonian PDEs is derived in the present paper, which is shown to be a discrete counterpart to that possessed by the differential equation. This provides further understanding of MSRK methods. However, much still remains to be investigated further.