Kth-Order Closed Form Difference Equations for 1-Dimensional, First and Second order, Linear Hyperbolic Equations, with Applications to Quasilinear Systems

We further investigate the high order positivity-preserving discontinuous Galerkin (DG) methods for linear hyperbolic and radiative transfer equations developed in Yuan et al. (SIAM J Sci Comput 38:A2987–A3019, 2016). The DG methods in Yuan et al. (2016) can maintain positivity and high order accuracy, but they rely both on the scaling limiter in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and a rotational limiter, the latter may alter cell averages of the unmodulated DG scheme, thereby affecting conservation. Even though a Lax–Wendroff type theorem is proved in Yuan et al. (2016), guaranteeing convergence to weak solutions with correct shock speed when such rotational limiter is applied, it would still be desirable if a conservative DG method without changing the cell averages can be obtained which has both high order accuracy and positivity-preserving capability. In this paper, we develop and analyze such a DG method for both linear hyperbolic equations and radiative transfer equations. In the one-dimensional case, the method uses traditional DG space $$P^k$$ of piecewise polynomials of degree at most k. A key result is proved that the unmodulated DG solver in this case can maintain positivity of the cell average if the inflow boundary value and the source term are both positive, therefore the positivity-preserving framework in Zhang and Shu (2010) can be used to obtain a high order conservative positivity-preserving DG scheme. Unfortunately, in two-dimensions this is no longer the case. We show that the unmodulated DG solver based either on $$P^k$$ or $$Q^k$$ spaces (piecewise kth degree polynomials or piecewise tensor-product kth degree polynomials) could generate negative cell averages. We augment the DG space with additional functions so that the positivity of cell averages from the unmodulated DG solver can be restored, thereby leading to high order conservative positivity-preserving DG scheme based on these augmented DG spaces following the framework in Zhang and Shu (2010). Computational results are provided to demonstrate the good performance of our DG schemes.

A mathematical model was developed to compute the motions of semisubmersible drilling vessels in waves for a wide variety of semisubmersible configurations. The model was derived from a linear representation of motions, ocean waves, and forces. The semisubmersible is represented as a rigid space frame composed of a number of cylindrical members with arbitrary diameters, lengths and orientations. Forces on the semisubmersible are derived from anchorline properties, and hydrostatic hydrodynamic principles. A solution is obtained for motions in six degrees of freedom for a sinusoidal wave train of arbitrary height, period, direction and water depth. Results from the analysis of three semisubmersibles are compared with results from available model test data to verily the mathematical model. Introduction An accurate and complete representation of the response of a drilling vessel to waves is a valuable engineering tool for predicting vessel performance and designing drilling equipment. The performance and designing drilling equipment. The wave response for a floating vessel may be obtained to various degrees of accuracy from model tests or analytical means, as described by Barkley and Korvin-Kroukovsky and as applied by Bain. A review of the works cited shows that the evaluation of the wave response for a particular vessel requires considerable time and effort, either in model construction and testing or in computer programming and calculations. In order to reduce programming and calculations. In order to reduce the amount of time and effort required to evaluate a particular vessel, means were investigated to generalize and automate, on a digital computer, methods for evaluating wave response for vessels of arbitrary configuration. The mathematical model described in this paper is the result of such an investigation for semisubmersible-type drilling vessels. The paper presents a general description of the mathematical model and the basic principles and assumptions from which it was derived. The validity of the model is evaluated by comparing results of the analysis of three semisubmersibles with available model test data. MATHEMATICAL MODEL The mathematical model for calculating the motions of a semisubmersible in waves is derived from basic principles and empirical relationships in classical mechanics. All equations are derived for "small amplitude" waves and motions. The nonlinear equations that appear in the problem are replaced by "equivalent" linear equations in order to conform to the linear analysis method used in obtaining a solution. The model is implemented in a computer program that computes vessel response in all six degrees of freedom for a broad range of semisubmersible configurations and wave parameters. The basic elements in the theoretical model are outlined, with a more detailed discussion of the principles and derivations used to obtain the model principles and derivations used to obtain the model presented in the Appendix. presented in the Appendix. SEMISUBMERSIBLE DESCRIPTION AND EQUATIONS OF MOTION The semisubmersible is characterized as a space-frame of cylindrical members and is described geometrically by specifying end-coordinates and diameters for all of the members. Specification of the mass, moments of inertia, center of gravity and floating position are required to complete the description. The six equations of motion for the semisubmersible derive from Newton's second law for a rigid body. These differential equations, when written in matrix form, equate the product of the six-component acceleration vector, {x}, and the inertia matrix, I, to a six-component, force-moment vector, {FT}. SPEJ P. 311

Solution of nonlinear partial differential equations from base equations

II.A - Exact Methods for ODEs

The Saint Venant equations are the equations governing open channel flow when a flood propagates. To obtain an analytical solution, the Saint Venant equations are approximately transformed to a nonlinear diffusion equation. The numerical result of the nonlinear diffusion equation explains the nonlinearity of the open channel flow and overland flow and the resulting asymmetry of the water surface profile. When the Saint Venant equations are numerically applied to flood routing in a finite region and finite time, a downstream boundary condition is necessary. Analytical solution of the appropriate nonlinear diffusion equation does not require the downstream boundary condition. The numerical result of the analytical solution of the approximate nonlinear diffusion equation is found to be suitable for simulation of the original nonlinear differential equation and the Saint Venant equations. The linearized diffusion model is found to be restricted in practical applications, and it cannot express nonlinea...

Here, first we shall set together various basic statements on the periodicity of the solutions of first order linear difference equations, then we shall define various discrete functions which are in a sense connected with the concept of periodicity. Finally, we will formulate a decomposition theorem for the solutions of first order linear difference equations with periodic coefficients.

PurposeThis paper aims to solve linear and non-linear shallow water wave equations using homotopy perturbation method (HPM). HPM is a straightforward method to handle linear and non-linear differential equations. As such here, one-dimensional shallow water wave equations have been considered to solve those by HPM. Interesting results are reported when the solutions of linear and non-linear equations are compared.Design/methodology/approachHPM was used in this study.FindingsSolution of one-dimensional linear and non-linear shallow water wave equations and comparison of linear and non-linear coupled shallow water waves from the results obtained using present method.Originality/valueCoupled non-linear shallow water wave equations are solved.

An example of a locally unsolvable hyperbolic equation of the second order is constructed, which has smooth ($C^∞$) coefficients, but has no solutions in the class of distributions.

As part of a program to improve understanding of the dynamics of the complicated, vigorous eddy and jet flow fields recently observed over the continental shelf and slope, we investigate the potential of intermediate models for use in both process and data assimilation studies of these flows. Intermediate models incorporate physics simpler than that contained in the full primitive equations yet more complete than in the quasi-geostrophic equations, and are capable of representing subinertial flows over O(1) bottom topographic variations and/or with O(1) isopycnal slopes. In addition, intermediate models dynamically filter out high-frequency gravity- inertial motions leading, potentially, to higher computational efficiency and well-posed limited area forecast/hindcast models. Initial studies focus on single layer flows on an f-plane with a free surface, governed by the shallow-water equations. In Part I, various intermediate models are formulated and their accuracy assessed by comparing some exact nonlinear analytical solutions that exist for the shallow-water equations with corresponding analytical solutions of the intermediate models. Here in Part II, an extensive set of numerical finite-difference solutions to initial-value problems in doubly periodic domains (to isolate model differences from the influence of boundary condition implementation on solid walls) is used to determine the accuracy of various intermediate models by comparing their predictions with those of a shallow-water equation model that uses a potential enstrophy and energy conserving numerical scheme (SWE). Intermediate model results are also contrasted with those from a quasi-geostrophic (QG) model. The intermediate models considered are based on the geostrophic momentum (GM) approximation, the derivation of Salmon utilizing Hamilton's principle (HP), a geostrophic vorticity (GV) approximation, a combination of the quasi-geostrophic momentum and full continuity equations (IM), the linear balance equations (LBE), the balance equations (BE), the related balance-type (HBE, BEM, NBE) and modified linear balance equations (LQBE), and on Lynch's slow equations in their original form (SE) and in a modified form (MSE). In addition, a semi-implicit version (SEMI) of the shallow-water equations, which numerically filters high-frequency motions, is included in the study. The basic initial-value problem used to test the various intermediate models involves sinusoidal flow over a symmetric Gaussian-shaped bottom topographic feature. Comparisons are made for a range of the relevant dimensionless model parameters including the strength of the flow (as measured by the Rossby number), the square of the ratio of a characteristic horizontal length scale to the Rossby radius of deformation, and the height of the topographic feature. A second initial-value problem involves the evolution of a rotating elliptical vortex over both flat and variable bottom topography. Results show that in cases with low local Rossby number flow, but with large topographic height, most of the intermediate models are substantially more accurate than QG. Even for flows with O(1) local Rossby numbers, some of the intermediate models continue to give excellent results. Specifically, BE, BEM and SEMI consistently do the best in the comparisons. Although the relative ordering of the remaining models is somewhat parameter-dependent, in general the next most accurate models are MSE and LQBE followed by HP and NBE. These are followed in quality by GV and GM while the remaining models, HBE, IM and LBE, perform least accurately for the range of parameters studied. Generally, intermediate models that have an integral invariant corresponding to conservation of potential enstrophy do better than those without, with the best results coming from models which have this property and a nonlinear balance equation.

STUDY OF HYPERBOLIC SYSTEM OF EQUATIONS TO STUDY TRAVEL TIME FOR TRAFFIC MODELLING by Anuj Nayyar 〈Dr. Hongtao Yang〉, Examination Committee Chair Associate Professor of Mathematical Sciences University of Nevada, Las Vegas In this thesis we study the travel time problem based on the known traffic density model. Using the conservation law, we model the travel time function by a boundary value problem of a non homogeneous linear hyperbolic equation. The equation is transformed into an initial value hyperbolic equation, and the well-posedness of the problem is discussed. The mathematical analysis for both density and travel problems are given. We also derive the analytic solutions for several special cases of traffic density. Numerical schemes are proposed for solving for travel time problem. Several numerical examples are presented and error analysis on the solutions obtained is performed to illustrate the rates of convergence of the numerical schemes.

Galerkin’s method for approximating the solutions of differential equations with Dirichlet boundary conditions requires the construction of finite-dimensional subspaces satisfying the boundary conditions. Except for special domains, this construction is difficult or impossible. Methods which avoid this difficulty are formulated. The main results are for nonlinear parabolic equations, for which a continuous scheme and several time-discrete schemes are formulated. Results are also obtained for linear elliptic and linear hyperbolic equations.

This article presents a new reliable solver based on polynomial approximation, using the Euler polynomials to construct the approximate solutions of the second-order linear hyperbolic partial differential equations with two variables and constant coefficients. Also, a formula expressing explicitly the Euler expansion coefficients of a function with one or two variables is proved. Another explicit formula, which expresses the two dimensional Euler operational matrix of differentiation is also given. Application of these formulae for reducing the problem to a system of linear algebraic equations with the unknown Euler coefficients, is explained. Hence, the result system can be solved and the unknown Euler coefficients can be found approximately. Illustrative examples with comparisons are given to confirm the reliability of the proposed method. The results show the efficiency and accuracy of the present work.

Modeling of tsunamis in glacial fjords prompts us to evaluate applicability of the cross‐sectionally averaged nonlinear shallow water equations to model propagation and runup of long waves in asymmetrical bays and also in fjords with two heads. We utilize the Tuck‐Hwang transformation, initially introduced for the plane beaches and currently generalized for bays with arbitrary cross section, to transform the nonlinear governing equations into a linear equation. The solution of the linearized equation describing the runup at the shore line is computed by taking into account the incident wave at the toe of the last sloping segment. We verify our predictions against direct numerical simulation of the 2‐D shallow water equations and show that our solution is valid both for bays with an asymmetric L‐shaped cross section, and for fjords with two heads—bays with a W‐shaped cross section.

Numerical investigations of fully nonlinear water waves generated by moving bottom topography

In this paper, we use quasilinearization technique, product integration rule, and collocation method to present a new numerical method to solve nonlinear fractional Volterra integro‐differential equations with logarithmic weakly singular kernel. After examining the behavior of the solution of the integro‐differential equation, we convert it into a nonlinear Volterra integral equation with a logarithmic weakly singular kernel. We implement the quasilinearization method on the resulting nonlinear equation and derive a sequence of linear integral equations with low‐smooth solutions that are convergent to the solution of the nonlinear equation. By utilizing a regularization technique, we enhance the smoothness of the solutions of the linear equations. Then, we apply collocation method along with the product integration rule, to transform the regularized linear equations into a linear algebraic system with a unique solution. We discuss the convergence of the presented method and estimate the error bounds for the proposed numerical approach. Finally, through several numerical examples, we practically test the effectiveness and accuracy of the proposed numerical method, where the obtained numerical results show their agreement with the discussed theoretical results.