Implicit-explicit multistep formulations for finite element discretisations using continuous interior penalty

We consider a finite element method with symmetric stabilisation for the discretisation of the transient convection--diffusion equation. For the time-discretisation we consider either the second order backwards differentiation formula or the Crank-Nicolson method. Both the convection term and the associated stabilisation are treated explicitly using an extrapolated approximate solution. We prove stability of the method and the $\tau^2 + h^{p+{\frac12}}$ error estimates for the $L^2$-norm under either the standard hyperbolic CFL condition, when piecewise affine ($p=1$) approximation is used, or in the case of finite element approximation of order $p \ge 1$, a stronger, so-called $4/3$-CFL, i.e. $\tau \leq C h^{4/3}$. The theory is illustrated with some numerical examples.

Numerical differentiation on scattered data through multivariate polynomial interpolation

We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor’s formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.

Some Characterizations of the Extended Beta and Gamma Functions: Properties and Applications

The article represents the elementary and general introduction of some characterizations of the extended gamma and beta Functions and their important properties with various representations. This paper provides reviews of some of the new proposals to extend the form of basic functions and some closed-form representation of more integral functions is described. Some of the relative behaviors of the extended function, the special cases resulting from them when fixing the parameters, the decomposition equation, the integrative representation of the proposed general formula, the correlations related to the proposed formula, the frequency relationships, and the differentiation equation for these basic functions were investigated. We also investigated the asymptotic behavior of some special cases, known formulas, the basic decomposition equation, integral representations, convolutions, recurrence relations, and differentiation formula for these target functions by studying. Applications of these functions have been presented in the evaluation of some reversible Laplace transforms to the complex of definite integrals and the infinite series of related basic functions.

Uniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equations

In this research, we developed a uniform order eleven of eight step Second derivative hybrid block backward differentiation formula for integration of stiff systems in ordinary differential equations. The single continuous formulation developed is evaluated at some grid point of x=x_(n+j),j=0,1,2,3,4,5 and6 and its first derivative was also evaluated at off-grid point x=x_(n+j),j=15/2 and grid point x=x_(n+j),j=8. The method is suitable for the solution of stiff ordinary differential equations and the accuracy and stability properties of the newly constructed method are investigated and are shown to be A-stable. Our numerical results obtained are compared with the theoretical solutions as well as ODE23 solver.

Simulations of nonlinear parabolic PDEs with forcing function without linearization

This paper aims at producing numerical solutions of nonlinear parabolic PDEs with forcing term without any linearization. Since the linearization of nonlinear term leads to lose real features, without doing linearization, this paper focuses on capturing natural behaviour of the mechanism. Therefore we concentrate on analysis of the physical processes without losing their properties. To carry out this study, a backward differentiation formula in time and a spline method in space have been combined in leading to the discretized equation. This method leads to a very reliable alternative in solving the problem by conserving the physical properties of the nature. The efficiency of the present method are proved theoretically and illustrated by various numerical tests.

DERIVATION OF 2-POINT ZERO STABLENUMERICAL ALGORITHM OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

This paper is aimed at deriving a 2-point zero stable numerical algorithm of block backward differentiation formula using Taylor series expansion, for solving first order ordinary differential equation. The order and zero stability of the method are investigated and the derived method is found to be zero stable and of order 3. Hence, the method is suitable for solving first order ordinary differential equation. Implementation of the method has been considered

ORDER AND CONVERGENCE OF THE ENHANCED 3-POINT FULLY IMPLICIT SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING INITIAL VALUE PROBLEM

This paper studied an enhanced 3-point fully implicit super class of block backward differentiation formula for solving stiff initial value problems developed by Abdullahi & Musa and go further to established the necessary and sufficient conditions for the convergence of the method. The method is zero stable, A-stable and it is of order 5. The method is found to be suitable for solving first order stiff initial value problems

Existence, uniqueness and regularity of the projection onto differentiable manifolds

We investigate the maximal open domain $${\mathscr {E}}(M)$$ E ( M ) on which the orthogonal projection map p onto a subset $$M\subseteq {{\mathbb {R}}}^d$$ M ⊆ R d can be defined and study essential properties of p. We prove that if M is a $$C^1$$ C 1 submanifold of $${{\mathbb {R}}}^d$$ R d satisfying a Lipschitz condition on the tangent spaces, then $${\mathscr {E}}(M)$$ E ( M ) can be described by a lower semi-continuous function, named frontier function. We show that this frontier function is continuous if M is $$C^2$$ C 2 or if the topological skeleton of $$M^c$$ M c is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a $$C^k$$ C k -submanifold M with $$k\ge 2$$ k ≥ 2 , the projection map is $$C^{k-1}$$ C k - 1 on $${\mathscr {E}}(M)$$ E ( M ) , and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order differentials on tubular neighborhoods. A sufficient condition for the inclusion $$M\subseteq {\mathscr {E}}(M)$$ M ⊆ E ( M ) is that M is a $$C^1$$ C 1 submanifold whose tangent spaces satisfy a local Lipschitz condition. We prove in a new way that this condition is also necessary. More precisely, if M is a topological submanifold with $$M\subseteq {\mathscr {E}}(M)$$ M ⊆ E ( M ) , then M must be $$C^1$$ C 1 and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between $${\mathscr {E}}(M)$$ E ( M ) and the topological skeleton of $$M^c$$ M c .

ENHANCED 3-POINT FULLY IMPLICIT SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING STIFF INITIAL VALUE PROBLEMS

This paper modified an existing 3–point block method for solving stiff initial value problems. The modification leads to the derivation of another 3 – point block method which is suitable for solving stiff initial value problems. The method approximates three solutions values per step and its order is 5. Different sets of formula can be generated from it by varying a parameter ρ ϵ (-1, 1) in the formula. It has been shown that the method is both Zero stable and A–Stable. Some linear and nonlinear stiff problems are solved and the result shows that the method outperformed an existing method and competes with others in terms of accuracy