A New Theoretical and Computational Framework for Computing Solution of Higher Classes With Application to Gasdynamics in Eulerian Frame of Reference

2001 ◽  
Author(s):  
K. S. Surana ◽  
Ali R. Ahmadi
Keyword(s):  
Differential Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Strong Solutions ◽  
Integral Form ◽  
Frame Of Reference ◽  
Mathematical Framework ◽  
Partial Differential ◽  
One Dimensional

Abstract This paper presents a new computational strategy along with a computational and mathematical framework for computing non-weak numerical solutions of stationary and time dependent partial differential equations. This approach utilizes strong form of the governing differential equations (GDE) and least squares approach in constructing the integral form. This new proposed approach is applied to one dimensional transient gasdynamics equation in Eulerian frame of reference using ρ, u, T as dependent variables. The currently used finite element approaches seek convergence of a solution in a fixed order space by h, p, or hp-adaptive processes. The fundamental point of departure in the proposed approach is that we seek the convergence of the computed converged solution over the spaces of different orders containing the basis functions. With this approach, dramatically higher convergence rates than those obtained for h, p, or hp-processes are achievable and the sequence of progressively converged solutions over the spaces of progressively increasing order in fact converge to the strong solution (analytical or theoretical) of the partial differential equation. It is demonstrated using one-dimensional transient Navier-Stokes equations for compressible fluid flow in Eulerian frame of reference, that in the presence of physical diffusion and dissipation, our computed solutions have exactly the same characteristics as the strong solutions. Riemann shock tube is used as a model problem.

Computations of Solutions of Higher Classes of Gasdynamics Equations in Lagrangian Frame of Reference Using a New Theoretical and Computational Framework

2001 ◽  
Author(s):  
K. S. Surana ◽  
Ali R. Ahmadi
Keyword(s):  
Differential Equations ◽  
Convergence Rates ◽  
Stokes Equations ◽  
Least Squares Method ◽  
Frame Of Reference ◽  
Navier Stokes ◽  
Partial Differential ◽  
Computational Framework ◽  
One Dimensional ◽  
Lagrangian Frame

Abstract In this paper we present and utilize a new theoretical and computational framework for computing solutions of higher classes of one dimensional transient Navier-Stokes partial differential equations in Lagrangian frame of reference using ρ, u, T variables. The approach utilizes ‘strong form’ of the governing differential equations (GDEs) and least squares method in constructing integral form. The currently used finite element approaches seek convergence of a solution in a fixed order space by h, p, or hp-adaptive processes. The fundamental point of departure in the proposed approach is that we seek the convergence of the computed converged solution over the spaces of different orders containing the basis functions. With this approach, dramatically higher convergence rates than those obtained for h, or hp-processes are achievable and the sequence of progressively converged solutions over the spaces of progressively increasing order in fact converge to the strong solution (analytical or theoretical) of the partial differential equation. It is demonstrated using one-dimensional transient Navier-Stokes equations for compressible fluid flow in Lagrangian frame of reference in ρ, u, T that in the presence of physical diffusion and dissipation, our computed converged solutions have exactly the same characteristics as the strong solutions. Compression of air in a rigid cylinder is used as the model problem.

Computations of Higher Class Solutions of Partial Differential Equations

2001 ◽  
Author(s):  
K. S. Surana ◽  
M. A. Bona
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Convergence Rates ◽  
Integral Form ◽  
Mathematical Framework ◽  
Partial Differential ◽  
Computational Framework ◽  
Fundamental Point ◽  
Adaptive Processes ◽  
Fixed Order

Abstract This paper presents a new computational strategy, computational framework and mathematical framework for numerical computations of higher class solutions of differential and partial differential equations. The approach presented here utilizes ‘strong forms’ of the governing differential equations (GDE’s) and least squares approach in constructing the integral form. The conventional, or currently used, approaches seek the convergence of a solution in a fixed (order) space by h, p or hp-adaptive processes. The fundamental point of departure in the proposed approach is that we seek convergence of the computed solution by changing the orders of the spaces of the basis functions. With this approach convergence rates much higher than those from h,p–processes are achievable and the progressively computed solutions converge to the ‘strong’ i.e. ‘theoretical’ solutions of the GDE’s. Many other benefits of this approach are discussed and demonstrated. Stationary and time-dependant convection-diffusion and Burgers equations are used as model problems.

scholarly journals A Sixth Order Accuracy Solution to a System of Nonlinear Differential Equations with Coupled Compact Method

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Don Liu ◽  
Qin Chen ◽  
Yifan Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Sixth Order ◽  
Compact Schemes ◽  
Order Convergence ◽  
Order Accuracy ◽  
Partial Differential

A system of coupled nonlinear partial differential equations with convective and dispersive terms was modified from Boussinesq-type equations. Through a special formulation, a system of nonlinear partial differential equations was solved alternately and explicitly in time without linearizing the nonlinearity. Coupled compact schemes of sixth order accuracy in space were developed to obtain numerical solutions. Within couple compact schemes, variables and their first and second derivatives were solved altogether. The sixth order accuracy in space is achieved with a memory-saving arrangement of state variables so that the linear system is banded instead of blocked. This facilitates solving very large systems. The efficiency, simplicity, and accuracy make this coupled compact method viable as variational and weighted residual methods. Results were compared with exact solutions which were obtained via devised forcing terms. Error analyses were carried out, and the sixth order convergence in space and second order convergence in time were demonstrated. Long time integration was also studied to show stability and error convergence rates.

The slow translation of a sphere in a rotating viscous fluid

1982 ◽  
Vol 117 ◽  
pp. 251-267 ◽  
Author(s):  
S. C. R. Dennis ◽  
D. B. Ingham ◽  
S. N. Singh
Keyword(s):  
Reynolds Number ◽  
Partial Differential Equations ◽  
Numerical Methods ◽  
Angular Velocity ◽  
Finite Difference ◽  
Differential Equations ◽  
Viscous Fluid ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Partial Differential

The motion of a sphere along the axis of rotation of an incompressible viscous fluid that is rotating as a solid mass is investigated by means of numerical methods for small values of the Reynolds and Taylor numbers. The Navier–Stokes equations governing the steady axisymmetric flow can be written as three coupled, nonlinear, elliptic partial differential equations for the stream function, vorticity and rotational velocity component. Two numerical methods are employed to solve these equations. The first is the method of series truncation in which the dependent variables are expressed as series of orthogonal Gegenbauer functions and the equations of motion are then reduced to three coupled sets of ordinary differential equations, which are integrated numerically subject to their boundary conditions. In the second method, specialized finite–difference techniques of solution are applied to the two-dimensional partial differential equations. These techniques employ finite-difference equations with coefficients that depend upon the exponential function; a particularly suitable form of approximation for use in calculating numerical solutions is obtained by expanding the exponential coefficients in powers of their exponents.Calculated results obtained by the two methods are in good agreement with each other. The calculations have been carried out according to theoretical assumptions that simulate the experiments of Maxworthy (1965) in which the sphere experiences no resultant torque exerted by the surrounding fluid and is free to rotate with constant angular velocity. Numerical estimates of this angular velocity and of the drag exerted by the fluid on the sphere are found to agree well with the experimental results for Reynolds and Taylor numbers in the range from zero to unity. The results for small values of the Reynolds number are also consistent with theoretical work of Childress (1963, 1964) which is valid as the Reynolds number tends to zero.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

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