983 research outputs found
Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients
open access articleMotivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge-Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations
Very High-Order A-stable Stiffly Accurate Diagonally Implicit Runge-Kutta Methods with Error Estimators
A numerical search approach is used to design high-order diagonally implicit
Runge-Kutta (DIRK) schemes equipped with embedded error estimators, some of
which have identical diagonal elements (SDIRK) and explicit first stage
(ESDIRK). In each of these classes, we present new A-stable schemes of order
six (the highest order of previously known A-stable DIRK-type schemes) up to
order eight. For each order, we include one scheme that is only A-stable as
well as schemes that are L-stable, stiffly accurate, and/or have stage order
two. The latter types require more stages, but give better convergence rates
for differential-algebraic equations (DAEs), and those which have stage order
two give better accuracy for moderately stiff problems. The development of the
eighth-order schemes requires, in addition to imposing A-stability, finding
highly accurate numerical solutions for a system of 200 equations in over 100
variables, which is accomplished via a combination of global and local
optimization strategies. The accuracy, stability, and adaptive stepsize control
of the schemes are demonstrated on diverse problems
Numerical investigations on global error estimation for ordinary differential equations
AbstractFour techniques of global error estimation, which are Richardson extrapolation (RS), Zadunaisky's technique (ZD), Solving for the Correction (SC) and Integration of Principal Error Equation (IPEE) have been compared in different integration codes (DOPRI5, DVODE, DSTEP). Theoretical aspects concerning their implementations and their orders are first given. Second, a comparison of them based on a large number of tests is presented. In terms of cost and precision, SC is a method of choice for one-step methods. It is much more precise and less costly than RS, and leads to the same precision as ZD for half its cost. IPEE can provide the order of the error for a cheap cost in codes based on one-step methods. In multistep codes, only RS and IPEE have been implemented since they are the only ones whose theoretical justification has been extended to this case. There, RS still provides a more reliable estimation than IPEE. However, as these techniques are based on variations of the global error, irrespective of the numerical method used, they fail to provide any more usefull information once the numerical method has reached its limit of accuracy due to the finite arithmetic
Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients
Motivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge–Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with a periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations
A computer program for the mixed analysis of variance model based on maximum likelihood
Computer program for mixed analysis of variance model based on maximum likelihoo
Embedded error estimation and adaptive step-size control for optimal explicit strong stability preserving Runge--Kutta methods
We construct a family of embedded pairs for optimal strong stability
preserving explicit Runge-Kutta methods of order to be used
to obtain numerical solution of spatially discretized hyperbolic PDEs. In this
construction, the goals include non-defective methods, large region of absolute
stability, and optimal error measurement as defined in [5,19]. The new family
of embedded pairs offer the ability for strong stability preserving (SSP)
methods to adapt by varying the step-size based on the local error estimation
while maintaining their inherent nonlinear stability properties. Through
several numerical experiments, we assess the overall effectiveness in terms of
precision versus work while also taking into consideration accuracy and
stability.Comment: 22 pages, 49 figure
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