Related papers: Higher order nonlocal operator method
We propose a nonlocal operator method for solving partial differential equations (PDEs). The nonlocal operator is derived from the Taylor series expansion of the unknown field, and can be regarded as the integral form "equivalent" to the…
This paper demonstrates new methods and implementations of nonlinear solvers with higher-order of convergence, which is achieved by efficiently computing higher-order derivatives. Instead of computing full derivatives, which could be…
In this study, we propose high-order implicit and semi-implicit schemes for solving ordinary differential equations (ODEs) based on Taylor series expansion. These methods are designed to handle stiff and non-stiff components within a…
Developments of nonlocal operators for modeling processes that traditionally have been described by local differential operators have been increasingly active during the last few years. One example is peridynamics for brittle materials and…
A development of an inverse first-order divided difference operator for functions of several variables is presented. Two generalized derivative-free algorithms builded up from Ostrowski's method for solving systems of nonlinear equations…
The space nonlocal Allen-Cahn equation is a famous example of fractional reaction-diffusion equations. It is also an extension of the classical Allen-Cahn equation, which is widely used in physics to describe the phenomenon of two-phase…
In this paper, we develop a regularized higher-order Taylor based method for solving composite (e.g., nonlinear least-squares) problems. At each iteration, we replace each smooth component of the objective function by a higher-order Taylor…
We present generalizations of Newton's method that incorporate derivatives of an arbitrary order $d$ but maintain a polynomial dependence on dimension in their cost per iteration. At each step, our $d^{\text{th}}$-order method uses…
In this work, we propose a high-order multiscale method for an elliptic model problem with rough and possibly highly oscillatory coefficients. Convergence rates of higher order are obtained using the regularity of the right-hand side only.…
We propose a systemic method of applying the auxiliary systems of original equations to find the high order nonlocal symmetries of nonlinear evolution equation. In order to validate the effectiveness of the method, some examples are…
Higher-order numerical methods are used to find accurate numerical solutions to hyperbolic partial differential equations and equations of transport type. Limiting is required to either converge to the correct type of solution or to adhere…
In this paper, we propose a third-order Newton's method which in each iteration solves a semidefinite program as a subproblem. Our approach is based on moving to the local minimum of the third-order Taylor expansion at each iteration,…
Higher-order tensor methods were recently proposed for minimizing smooth convex and nonconvex functions. Higher-order algorithms accelerate the convergence of the classical first-order methods thanks to the higher-order derivatives used in…
This paper proposes a higher-order multiscale computational method for nonlinear thermo-electric coupling problems of composite structures, which possess temperature-dependent material properties and nonlinear Joule heating. The innovative…
We present a novel approach for high-order accurate numerical differentiation on unstructured meshes of quadrilateral elements. To differentiate a given function, an auxiliary function with greater smoothness properties is defined which…
We systematically introduce the idea of applying differential operator method to find a particular solution of an ordinary nonhomogeneous linear differential equation with constant coefficients when the nonhomogeneous term is a polynomial…
In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the…
We introduce basic aspects of new operator method, which is very suitable for practical solving differential equations of various types. The main advantage of the method is revealed in opportunity to find compact exact operator solutions of…
We consider the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients (methods with real coefficients of order greater than two necessarily have negative…
In this paper, a centred universal high-order finite volume method for solving hyperbolic balance laws is presented. The scheme belongs to the family of ADER methods where the Generalized Riemann Problems (GRP) is a building block. The…