Decoupled Direct State Transition Matrix Calculation with Runge-Kutta Methods

The main aim of this paper was to find the correct method of calculating equations of heat and mass transfer for the adsorption process and to calculate it numerically in reasonable time and with proper accuracy. An adsorption heat pump with a silica gel adsorbent and water adsorbate is discussed. We developed a mathematical model of temperature and uptake changes in the adsorber/desorber comprising the set of heat and mass balance partial differential equations (PDEs), together with the initial and boundary conditions and solved it by the numerical method of lines (NMOL). Spatial discretization was performed with equally spaced axial nodes and the PDEs were reduced to a set of ordinary differential equations (ODEs). We focused on the comparison of results obtained when the set of heat and mass balance ODEs for an adsorber was solved using: (1) the Runge–Kutta fixed step size fourth-order method (RKfixed), (2) the Runge–Kutta–Fehlberg 4.5th-order method with a variable step size (RK45), and (3) the Gear Backward Differentiation Formulae numerical (Gear BDF) methods. In our experience, all three types of ODE numerical methods (RKfixed, RK45, and Gear BDF) can be applied in simple models to model an adsorber with attention on their limitations. The Gear BDF method usually requires much fewer steps than the RK45 method for almost the same calculating time. RK methods require many more steps to obtain results, and the calculating time depends on accuracy or defined time step. Moreover, one should pay attention to the number of nodes or possible oscillations.

In this work, a novel quantum Fourier ordinary differential equation (ODE) solver is proposed to solve both linear and nonlinear partial differential equations (PDEs). Traditional quantum ODE solvers transform a PDE into an ODE system via spatial discretization and then integrate it, thereby converting the task of solving the PDE into computing the integral for the driving function f(x). These solvers rely on the quantum amplitude estimation algorithm, which requires the driving function f(x) to be within the range of [0, 1] and necessitates the construction of a quantum circuit for the oracle R that encodes f(x). This construction can be highly complex, even for simple functions like f(x) = x. An important exception arises for the specific case of f(x) = sin2(mx + c), which can be encoded more efficiently using a set of Ry rotation gates. To address these challenges, we expand the driving function f(x) as a Fourier series and propose the quantum Fourier ODE Solver. This approach not only simplifies the construction of the oracle R but also removes the restriction that f(x) must lie within [0,1]. The proposed method was evaluated by solving several representative linear and nonlinear PDEs, including the Navier–Stokes (N–S) equations. The results show that the quantum Fourier ODE solver produces results that closely match both analytical and reference solutions.

Solving ordinary differential equations (ODEs) is vital in diverse fields. However, it is difficult to obtain the exact analytical solutions of ODEs due to their changeable mathematical forms. Traditional numerical methods can find approximate solutions for specific ODEs. Unfortunately, they often suffer from ODEs' forms and characteristics. To approximate different types of ODEs, this paper proposes a generic method based on adaptive differential evolution. Besides, in order to further reduce the error of the obtained approximate solutions, an improved Fourier periodic expansion function is developed, which is then combined with the least square weight method to formulate the ODEs as an optimization problem. Since the proposed method is not limited to ODEs' forms and constraint conditions, it can be used to approximate any ODEs, including linear ODEs and nonlinear ODEs. The proposed method is evaluated on twenty popular test cases. The results indicate that the proposed method is able to accurately approximate different ODEs with better performance compared with other methods.

The impact of a dam-break surge on a wall has been numerically modeled in this work. The incompressible and inviscid surge is advancing over a dry bed and the air entrapment has not been considered in the equations describing the phenomenon. The global quantities of interest has been evaluated in the framework of the potential theory. The numerical method couples a potential flow solver with an ordinary differential equation (ODE) solver. The potential flow solver is an implementation of a Mixed Finite Element method (MFE). A second order CrankNicholson scheme is used as ODE solver. The MFE solver turns out to be second order accurate for both the potential and the flow field and thls is the main advantage as compared to the corresponding finite volume scheme. The method has been used for the calculation of the 2-D unsteady, incompressible, free surface impact of clear water bores on vertical plane walls. Due to the accuracy of the scheme, a good agreement with the experimental data is obtained. A liquid-liquid impact has also been attempted.

The dynamic modeling and experimental validation of the model response of the poly(methyl methacrylate) (PMMA) cell-cast process for plastic sheet production is presented. It is based on the straightforward initiation, propagation, and termination reaction polymerization mechanism. The sheet molding process was modeled by a two-dimensional dynamic mathematical model able to predict conversion, temperature, and molecular weight averages. The mathematical model is cast as a partial differential equations (PDEs) system that is discretized using the numerical method of lines. The resulting set of ordinary differential equations, representing the heat and mass balances for this polymerization system, are then solved by standard ordinary differential equations (ODE) solvers. Comparison of the model prediction capabilities against experimental temperature measurements taken at the extremes of the PMMA sheet being produced are presented.

Implicit Runge–Kutta (IRK) methods are highly effective for solving stiff ordinary differential equations (ODEs) but can be computationally expensive for large-scale problems due to the need of solving coupled algebraic equations at each step. This study improves IRK efficiency by leveraging parallelism to decouple stage computations and reduce communication overhead, specifically we stably decouple a perturbed version of the stage system of equations and recover the exact solution by solving a Sylvester matrix equation with an explicitly known low-rank right-hand side. Two IRK families—symmetric methods and collocation methods—are analyzed, with extensions to nonlinear problems using a simplified Newton method. Implementation details, shared memory parallel code, and numerical examples, particularly for ODEs from spatially discretized PDEs, demonstrate the efficiency of the proposed IRK technique.

Parallel implementation for the two-stage SDIRK methods via diagonalization

We propose to formulate MRI image reconstruction as an optimization problem and model the optimization trajectory as a dynamic process using ordinary differential equations (ODEs). We model the dynamics in ODE with a neural network and solve the desired ODE with the off-the-shelf (fixed) solver to obtain reconstructed images. We extend this model and incorporate the knowledge of off-the-shelf ODE solvers into the network design (learned solvers). We investigate several models based on three ODE solvers and compare models with fixed solvers and learned solvers. Our models achieve better reconstruction results and are more parameter efficient than other popular methods such as UNet and cascaded CNN. We introduce a new way of tackling the MRI reconstruction problem by modeling the continuous optimization dynamics using neural ODEs.

In this paper, a new class of two-derivative two-step Runge-Kutta (TDTSRK) methods for the numerical solution of non-stiff initial value problems (IVPs) in ordinary differential equation (ODEs) is considered. The TDTSRK methods are a special case of multi-derivative Runge-Kutta methods proposed by Kastlunger and Wanner (1972). The methods considered herein incorporate only the first and second derivatives terms of ODEs. These methods possess large interval of stability when compared with other existing methods in the literature. The experiments have been performed on standard problems, and comparisons were made with some standard explicit Runge-Kutta methods in the literature.

The calculus of differential forms is used to discuss the local integration theory of a general set of autonomous first order ordinary differential equations. Geometrically, such a set is a vector field ${\bf V}$ in the space of dependent variables. Integration consists of seeking associated geometric structures invariant along ${\bf V}$: scalar fields (first integrals), forms (such as symplectic tensors), vectors (CT generators, Lie’s invariance generators), integrals over subspaces (which are not limited to Poincaré’s absolute and relative invariants). Most classical results are easily understood as simple relations between these. It is shown first that to any field ${\bf V}$ can be associated a Hamiltonian structure of forms (if, when dealing with an odd number of dependent variables, an arbitrary equation of constraint is also added). Families of integral invariants are an immediate consequence. Poisson brackets are isomorphic to Lie products of associated CT-generating vector fields. Hamilton’s variational principle follows from the fact that the maximal regular integral manifolds of a closed set of forms (here a Hamiltonian structure) must include the characteristics of the set (here the trajectories of ${\bf V}$). Auxiliary vector fields are also carefully discussed: the hierarchy of invariance generators of (i) the vector field ${\bf V}$, of (ii) its trajectories, of (iii) any associated Hamiltonian structure of forms. An Appendix summarizes the manipulation of vectors and forms, as a tool for applied mathematics.

Three-dimensional static analysis of Levy-type functionally graded plate with in-plane stiffness variation

Abstract—To construct a scheme of implicit Runge-Kutta methods, there are a number of coefficients that must be determined and satisfying consistency properties and Butcher’s simplifying assumptions. In this paper we provide the numerical simulation technique to obtain a scheme of 10th order Implicit Runge-Kutta (IRK10) method. For simulation process, we construct an algorithm to compute all the coefficients involved in the IRK10 scheme. The algorithm is implemented in a language programming (Turbo Pascal) to obtain all the required coefficients in the scheme. To show that our scheme works correctly, we use the scheme to solve Hénon-Heiles system.Keywords—ODEs, 10th order IRK method, numerical technique, Hénon-Heiles system

This research paper deals with a numerical algorithm based on the RCW method to solve a set of first-order ordinary differential equations. In this algorithm, the answer of each equation is considered as a polynomial of degree 2. The coefficients used in this polynomial are categorized into groups of free and fixed coefficients. The free coefficients are obtained by optimizing the error function, while the fixed coefficients are computed from the magnitudes of derivative of this polynomial at initial value. To check the correctness of the current algorithm, its results for several case problems are compared with the findings of the Runge-Kutta (RK) method. This comparison shows that the algorithm presented in this paper has a higher accuracy and stability than the RK method even for the nonlinear problems.

In the previous chapter, we introduced explicit Runge-Kutta (ERK) methods and demonstrated how they can be implemented as a hierarchy of Python classes. For most ODE systems, replacing the simple forward Euler method with a higher-order ERK method will significantly reduce the number of time steps needed to reach a specified accuracy. Furthermore, it often leads to reduced computation time, since the additional cost per time step is outweighed by the reduced number of steps. However, there exists a class of ODEs known as stiff systems, where all the ERK methods require very small time steps, and any attempt to increase the time step leads to spurious oscillations and possible divergence of the solution. Stiff ODE systems pose a challenge for explicit methods, and they are better addressed by implicit solvers such as implicit Runge-Kutta (IRK) methods. IRK methods are well-suited for stiff problems and can offer substantial reductions in computation time when tackling challenging problems.

Both Runge-Kutta and linear multistep methods are available to solve initial value problems for ordinary differential equations in the R packages deSolve and deTestSet. Nearly all of these solvers use adaptive step size control, some also control the order of the formula adaptively, or switch between different types of methods, depending on the local properties of the equations to be solved. We show how to trigger the various methods using a variety of applications pointing, where necessary, to problems that may arise. For instance, many practical applications involve discontinuities. As the integration routines assume that a solution is sufficiently differentiable over a time step, handing such discontinuities requires special consideration. We give examples of how we can implement a nonsmooth forcing term, switching behavior, and problems that include sudden jumps in the dependent variables. Since much computational efficiency can be gained by using the correct method for a particular problem, we end this chapter by providing a few guidelines as to how the most efficient solution method for a particular problem can be found.