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High-order methods beyond the classical complexity bounds: inexact high-order proximal-point methods
We introduce a Bi-level OPTimization (BiOPT) framework for minimizing the sum of two convex functions, where one of them is smooth enough. The BiOPT...
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On complexity and convergence of high-order coordinate descent algorithms for smooth nonconvex box-constrained minimization
Coordinate descent methods have considerable impact in global optimization because global (or, at least, almost global) minimization is affordable...
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Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High Order
We propose a new paradigm for designing efficient p -adaptive arbitrary high-order methods. We consider arbitrary high-order iterative schemes that...
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High Order ADER-IPDG Methods for the Unsteady Advection-Diffusion Equation
We present a high-order Galerkin method in both space and time for the 1D unsteady linear advection-diffusion equation. Three Interior Penalty...
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Arbitrary High Order ADER-DG Method with Local DG Predictor for Solutions of Initial Value Problems for Systems of First-Order Ordinary Differential Equations
An adaptation of the arbitrary high order ADER-DG numerical method with local DG predictor for solving the IVP for a first-order non-linear ODE...
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High Order Absolutely Convergent Fast Swee** Methods with Multi-resolution WENO Local Solvers for Eikonal and Factored Eikonal Equations
Fast swee** methods are a class of efficient iterative methods developed in the literature to solve steady-state solutions of hyperbolic partial...
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On the maximum principle and high-order, delay-free integrators for the viscous Cahn–Hilliard equation
The stabilization approach has been known to permit large time-step sizes while maintaining stability. However, it may “slow down the convergence...
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Digital Convergence
High-Performance Computing has recently been challenged by the advent of Artificial Intelligence. Artificial Intelligence has become rather popular...
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Wilton Ripples with High-Order Resonances in Weakly Nonlinear Models
Resonant periodic traveling waves (Wilton ripples) are examined asymptotically for a family of weakly nonlinear partial differential equations....
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Generalized high-order compact difference schemes for the generalized Rosenau–Burgers equation
A shallow-water wave propagation model can be described as a generalized Rosenau–Burgers equation with strong nonlinearity and high-order dispersion...
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Actual Accuracy of Linear Schemes of High-Order Approximation in Gasdynamic Simulations
AbstractA new test problem for one-dimensional gas dynamics equations is considered. Initial data in the problem is a periodic smooth wave. Shock...
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Curvilinear Mesh Generation for the High-Order Virtual Element Method (VEM)
We present a proof-of-concept methodology for generating curvilinear polygonal meshes suitable for high-order discretizations by the Virtual Element...
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A Hybrid High-Order Method for a Class of Strongly Nonlinear Elliptic Boundary Value Problems
In this article, we design and analyze a hybrid high-order (HHO) finite element approximation for a class of strongly nonlinear boundary value...
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Structure-Preserving Combined High-Order Compact Schemes for Multiple Order Spatial Derivatives Differential Equations
For differential equations with multiple order spatial derivatives, there are some shortcomings by the classical high order compact (HOC)...
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The Compact and Accuracy Preserving Limiter for High-Order Finite Volume Schemes Solving Compressible Flows
In this paper, a new limiter namely the compact weighted biased average procedure (CWBAP) is proposed for the shock capturing of high-order finite...
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High-order finite difference methods
Finite difference (FD) methods are very popular for solving partial differential equations (PDEs) because of their simplicity. A simple but powerful...
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High Order Numerical Scheme for Generalized Fractional Diffusion Equations
In this paper, a higher order finite difference scheme is proposed for generalized fractional diffusion equations (GFDEs). The fractional diffusion...
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High Order Asymptotic Preserving and Classical Semi-implicit RK Schemes for the Euler–Poisson System in the Quasineutral Limit
In this paper, the design and analysis of high order accurate IMEX finite volume schemes for the compressible Euler–Poisson (EP) equations in the...
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On Two Competing Methods with Optimal Eighth Order Convergence
High convergence order methods are important in computational mathematics, since they generate sequences converging to a solution of a non-linear...
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Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann Transport
We introduce an hp -version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new...
