ODEs In Time Research Articles

Modal Discontinuous Galerkin Simulations of Richtmyer–Meshkov Instability at Backward-Triangular Bubbles: Insights and Analysis

This paper investigates the dynamics of Richtmyer–Meshkov instability (RMI) in shocked backward-triangular bubbles through numerical simulations. Two distinct gases, He and SF6, are used within the backward-triangular bubble, surrounded by N2 gas. Simulations are conducted at two distinct strengths of incident shock wave, including Ms=1.25 and 1.50. A third-order modal discontinuous Galerkin (DG) scheme is applied to simulate a physical conservation laws of two-component gas flows in compressible inviscid framework. Hierarchical Legendre modal polynomials are employed for spatial discretization in the DG platform. This scheme reduces the conservation laws into a semi-discrete set of ODEs in time, which is then solved using an explicit 3rd-order SSP Runge–Kutta scheme. The results reveal significant effects of bubble density and Mach numbers on the growth of RMI in the shocked backward-triangular bubble, a phenomenon not previously reported. These effects greatly influence flow patterns, leading to intricate wave formations, shock focusing, jet generation, and interface distortion. Additionally, a detailed analysis elucidates the mechanisms driving vorticity formation during the interaction process. The study also thoroughly examines these effects on the flow fields based on various integral quantities and interface characteristics.

Numerical Simulation of Nonlinear and Non-Isothermal Liquid Chromatography for Studying Thermal Variations in Columns Packed with Core-Shell Particles

A high-resolution flux-limiting semi-discrete finite volume scheme (HR-FVS) is applied in this study to numerically approximate the nonlinear and non-isothermal flow of one-dimensional lumped kinetic model (1D-LKM), for a fixed-bed column loaded with core-shell particles. The developed model comprise a system of convection-dominated partial differential for mass and energy balances in the mobile phases coupled with differential equation and algebraic equation in the stationary phase. The solution of the model equations is obtained by utilizing a HR-FVS, the scheme has second-order accuracy even on the grid coarse and its explicit nature has the potential to resolve the arisen sharp discontinuities in the solution profiles. A second-order total variation diminishing (TVD) Runge-Kutta technique is used to solve the system of ODEs in time. Several forms of a single-solute mixture are produced to investigate the influences of the fractions of core radius on thermal waves and concentration fronts. Moreover, a particular criterion is introduced for analyzing the performance of the underlying process and to identify the optimal parameter values of the fraction of core radius.

A waveform relaxation Newmark method for structural dynamics problems

In the conventional Newmark family for time integration of hyperbolic problems, both explicit and implicit methods are inherently sequential in the time domain and not well suited for parallel implementations due to unavoidable processor communication at every time step. In this work we propose a Waveform Relaxation Newmark (WRN $$_\beta $$ ) algorithm for the solution of linear second-order hyperbolic systems of ODEs in time, which retains the unconditional stability of the implicit Newmark scheme with the advantage of the lower computational cost of explicit time integration schemes. This method is unstructured in the time domain and is well suited for parallel implementation. We consider a Jacobi and Gauss–Seidel type splitting and study their convergence and stability. The performance of the WRN $$_\beta $$ algorithm is compared to a standard implicit Newmark method and the obtained results confirm the effectiveness of the Waveform Relaxation Newmark algorithm as a new class of more efficient time integrators, which is applicable, as shown in the numerical examples, to both the finite element method and meshfree methods (e.g. the reproducing kernel particle method).

Efficient Semi-Discretization Techniques for Pricing European and American Basket Options

This paper studies the valuation of option pricing problem in the presence of several assets. European- and American-style options are dealt with using high order semi-discretization techniques. We integrate the resulting system of semi-discretized ODEs in time with several time-stepping schemes such as explicit Euler method, explicit Runge–Kutta method, and the Runge–Kutta method with adaptive variable step sizes. Numerical results demonstrate that employing higher order semi-discretizations improves the computational performance of the multi-asset option pricing problems.

Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems

The aim of this paper is to examine pattern formation in the sub— and super-diffusive scenarios and compare it with that of classical or standard diffusive processes in two-component fractional reaction-diffusion systems that modeled a predator-prey dynamics. The focus of the work concentrates on the use of two separate mathematical techniques, we formulate a Fourier spectral discretization method as an efficient alternative technique to solve fractional reaction-diffusion problems in higher-dimensional space, and later advance the resulting systems of ODEs in time with the adaptive exponential time-differencing solver. Obviously, the fractional Fourier approach is able to achieve spectral convergence up to machine precision regardless of the fractional order α, owing to the fact that our approach is able to give full diagonal representation of the fractional operator. The complexity of the dynamics in this system is theoretically discussed and graphically displayed with some examples and numerical simulations in one, two and three dimensions.

An efficient collocation algorithm for multidimensional wave type equations with nonlocal conservation conditions

In this paper, we derive and analyze an efficient spectral collocation algorithm to solve numerically some wave equations subject to initial-boundary nonlocal conservation conditions in one and two space dimensions. The Legendre pseudospectral approximation is investigated for spatial approximation of the wave equations. The Legendre–Gauss–Lobatto quadrature rule is established to treat the nonlocal conservation conditions, and then the problem with its nonlocal conservation conditions are reduced to a system of ODEs in time. As a theoretical result, we study the convergence of the solution for the one-dimensional case. In addition, the proposed method is extended successfully to the two-dimensional case. Several numerical examples with comparisons are given. The computational results indicate that the proposed method is more accurate than finite difference method, the method of lines and spline collocation approach.

A numerical study of two dimensional hyperbolic telegraph equation by modified B-spline differential quadrature method

The present paper uses a relatively new approach and methodology to solve second order two dimensional hyperbolic telegraph equation numerically. We use modified cubic B-spline basis functions based differential quadrature method for space discretization that reduces the problem into an amenable system of ordinary differential equations. The resulting system of ODEs in time subsequently have been solved by SSP-RK43 scheme. Stability of the scheme is studied using matrix stability analysis and found to be stable. The efficacy of proposed approach has been confirmed with seven numerical experiments, where comparison is made with some earlier work. It is clear that the results obtained are acceptable and are in good agreement with earlier studies. However, we obtain these results in much less CPU time. The method is very simple, efficient and produces very accurate numerical results in considerably smaller number of nodes and hence saves computational effort.

Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs

The present study considers mathematical classification of the time differential operators and then applies methods of approximation in time such as Galerkin method (GM ), Galerkin method with weak form (GM / WF ), Petrov-Galerkin method ( PGM), weighted residual method (WRY ), and least squares method or process ( LSM or LSP ) to construct finite element approximations in time. A correspondence is established between these integral forms and the elements of the calculus of variations: 1) to determine which methods of approximation yield unconditionally stable (variationally consistent integral forms, VC ) computational processes for which types of operators and, 2) to establish which integral forms do not yield unconditionally stable computations (variationally inconsistent integral forms, VIC ). It is shown that variationally consistent time integral forms in hpk framework yield computational processes for ODEs in time that are unconditionally stable, provide a mechanism of higher order global differentiability approximations as well as higher degree local approximations in time, provide control over approximation error when used as a time marching process and can indeed yield time accurate solutions of the evolution. Numerical studies are presented using standard model problems from the literature and the results are compared with Wilson’s θ method as well as Newmark method to demonstrate highly meritorious features of the proposed methodology.

A fluid-like model of vibrated granular layers: Linear stability, kinks, and oscillons

A continuum model is proposed for the dynamics of shallow non-cohesive granular layers driven by vertical vibration of a horizontal rigid plate in vacuo. The granular mass is assumed to behave as a Newtonian liquid while in flight and to exhibit a solid-like rebound from the vibrating plate. The simple version explored here involves two constant parameters, a kinematic viscosity and a coefficient of restitution. According to the present model, the sensitive dependence of the contact dynamics on layer shape represents the primary source of instability and pattern formation, since the lack of cohesive stress rules out the Rayleigh–Taylor instability driving Faraday patterns on liquid layers. This is borne out by a linear stability analysis which indicates that flat granular layers are linearly stable against free-surface perturbations and, hence, that the instability of vibrated granular layers is nonlinear in origin. An effort is made to capture certain aspects of the high Reynolds number nonlinear dynamics, namely spatially localized “kinks” and “oscillons”, based on purely rectilinear (“antiplane”) motion with constant layer thickness. Three different numerical methods were employed (1) computation of spatial amplitudes in a time-periodic solutions, (2) discretization of the underlying PDEs, with the resulting ODEs in time treated by an event-detecting integrator, and (3) a variant of the latter based on discretization of a well-known Green’s function. Inaccuracies in contact detection appear to give disagreements between the different methods, such that Method (2) produced stable f / 2 kinks and oscillons only if the variant (3) was employed, while Method (1) gave rise to stable oscillons but apparently unstable kinks. Also, while qualitatively similar, the numerically simulated oscillons exhibit differences from those observed experimentally. The present findings call for further work on the numerical methods, investigation of more complex solutions with lateral motion and varicose layers, and improvements in the constitutive model for the granular layer, such as the incorporation of a velocity-dependent restitution.

Pricing Multi-Asset American Options: A Finite Element Method-of-Lines with Smooth Penalty

This paper studies the problem of pricing multi-asset American-style options in the Black---Scholes---Merton framework. The value function of an option contract is known to satisfy a partial differential variational inequality (PDVI) when early exercise is permitted. We develop a computational method for the valuation of multi-asset American-style options based on approximating the PDVI by a non-linear penalized PDE with a penalty term with continuous Jacobian. We convert the non-linear PDE to a variational (weak) form, discretize the weak formulation spatially by a Galerkin finite element method to obtain a system of ODEs, and integrate the resulting system of ODEs in time with an adaptive variable order and variable step size solver SUNDIALS. Numerical results demonstrate that employing a penalty term with continuous Jacobian in contrast to the penalty terms with discontinuous Jacobians in use in the literature improves computational performance of the adaptive temporal integrator. In our framework we are able to price American-style options with payoffs dependent on up to six assets on a PC. This is in contrast to the existing literature on the pricing of American options by PDE methods, that has so far been limited to at most three-dimensional problems. Our results open avenues for further applications to multi-dimensional problems, such as pricing convertible bonds in multi-factor models, that will be explored in future work.

THE K-VERSION OF FINITE ELEMENT METHOD FOR NONLINEAR OPERATORS IN BVP

In the companion papers [1,2], authors introduced the concepts of k-version of finite element method and k, hk, pk, hkp-processes of the finite element method for boundary value problems described by self-adjoint and non-self adjoint operators using Ĥk,p(Ω) spaces with specific details including numerical studies for weak forms and least square processes. It was demonstrated that a variationally consistent (VC) weak form is possible when the differential operator is self-adjoint, however, in case of non-self-adjoint operators the weak forms are variationally inconsistent (VIC) which lead to degenerate computational processes that can produce spurious oscillations in the computed solutions. In this paper we demonstrate that when the boundary value problems are described by non-linear differential operators, Galerkin processes and weak forms can never be variationally consistent and hence result in degenerate computational processes and suffer from same problems as in the case of non-self-adjoint operators ...

THE k-VERSION OF FINITE ELEMENT METHOD FOR NON-SELF-ADJOINT OPERATORS IN BVP

THE k-VERSION OF FINITE ELEMENT METHOD FOR NON-SELF-ADJOINT OPERATORS IN BVP

THE k-VERSION OF FINITE ELEMENT METHOD FOR SELF-ADJOINT OPERATORS IN BVP

In this paper a new mathematical and computational framework for boundary value problems described by self-adjoint differential operators is presented. In this framework, numerically computed solutions, when converged, possess the same degree of global smoothness in terms of differentiability up to any desired order as the theoretical solutions. This is accomplished using spaces Ĥk,p that contain basis functions of degree p and order k - 1 (or the order of the space k). It is shown that the order of space k is an intrinsically important independent parameter in all finite element computational processes in addition to the discretization characteristic length h and the degree of basis functions p when the theoretical solutions are analytic. Thus, in all finite element computations, all quantities of interest (e.g., quadratic functional, error or residual functional, norms and seminorms, error norms, etc.) are dependent on h, p as well as k. Therefore, for fixed h and p, convergence of the finite element process can also be investigated by changing k, hence k-convergence and thus the k-version of finite element method. With h, p, and k as three independent parameters influencing all finite element processes, we now have k, hk, pk, and hpk versions of finite element methods. The issue of minimally conforming finite element spaces is reexamined and it is demonstrated that the definition of currently believed minimally conforming space which permit weak convergence of the highest-order derivatives of the dependent variables appearing in the bilinear form is not justifiable mathematically or from physics view point. A new criterion is proposed for establishing the minimally conforming spaces which is more in agreement with the physics and mathematics of the BVP. Significant features and merits of the proposed mathematical and computational framework are presented, discussed, illustrated, and substantiated mathematically as well as numerically with the Galerkin and least-squares finite element formulations for self-adjoint boundary-value problems.

Multi-symplectic spectral discretizations for the Zakharov–Kuznetsov and shallow water equations

The time evolution of multi-symplectic PDEs with periodic boundary conditions in one and two space dimensions is considered. We introduce the idea of a multi-symplectic Fourier transformfor systems on periodic domains leading to a semi-discretization on Fourier space, and to the concept of multi-symplecticity on Fourier space. The spatial discretization leads to a discrete lattice model with certain discrete conservation laws imposed on it — a discrete wave model. A by-product of the semi-discretization is that it leads automatically to a system of Hamiltonian ODEs in time when truncated. We show that the one-dimensional shallow water equations and the two-dimensional Zakharov–Kuznetsov equation are multi-symplectic and derive spectral discretizations for these systems and present numerical experiments.