Solution of Time-dependent Advection-Diffusion Problems with the Sparse-grid Combination Technique and a Rosenbrock Solver

Abstract In the current paper the efficiency of the sparse-grid combination tech- nique applied to time-dependent advection-diffusion problems is investigated. For the time-integration we employ a third-order Rosenbrock scheme implemented with adap- tive step-size control and approximate matrix factorization. Two model problems are considered, a scalar 2D linear, constant-coe±cient problem and a system of 2D non- linear Burgers' equations. In short, the combination technique proved more efficient than a single grid approach for the simpler linear problem. For the Burgers' equations this gain in efficiency was only observed if one of the two solution components was set to zero, which makes the problem more grid-aligned.

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Geometrically exact time-integration mesh-free schemes for advection-diffusion problems derived from optimal transportation theory and their connection with particle methods

International Journal for Numerical Methods in Engineering ◽

10.1002/nme.5552 ◽

2017 ◽

Vol 112 (9) ◽

pp. 1175-1193 ◽

Cited By ~ 3

Author(s):

L. Fedeli ◽

A. Pandolfi ◽

M. Ortiz

Keyword(s):

Time Integration ◽

Optimal Transportation ◽

Particle Methods ◽

Exact Time ◽

Advection Diffusion ◽

Mesh Free ◽

Transportation Theory ◽

Diffusion Problems

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Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics

Communications on Applied Mathematics and Computation ◽

10.1007/s42967-021-00159-w ◽

2021 ◽

Author(s):

Hendrik Ranocha ◽

Lisandro Dalcin ◽

Matteo Parsani ◽

David I. Ketcheson

Keyword(s):

Fluid Dynamics ◽

Error Control ◽

Time Integration ◽

Size Control ◽

Spectral Element ◽

Step Size ◽

Step Size Control ◽

Wide Range ◽

Cfd Applications ◽

Runge Kutta

AbstractWe develop error-control based time integration algorithms for compressible fluid dynamics (CFD) applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime. Focusing on discontinuous spectral element semidiscretizations, we design new controllers for existing methods and for some new embedded Runge-Kutta pairs. We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice. We compare a wide range of error-control-based methods, along with the common approach in which step size control is based on the Courant-Friedrichs-Lewy (CFL) number. The optimized methods give improved performance and naturally adopt a step size close to the maximum stable CFL number at loose tolerances, while additionally providing control of the temporal error at tighter tolerances. The numerical examples include challenging industrial CFD applications.

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Space–time discontinuous Galerkin method for advection–diffusion problems on time-dependent domains

Applied Numerical Mathematics ◽

10.1016/j.apnum.2005.11.003 ◽

2006 ◽

Vol 56 (12) ◽

pp. 1491-1518 ◽

Cited By ~ 44

Author(s):

J.J. Sudirham ◽

J.J.W. van der Vegt ◽

R.M.J. van Damme

Keyword(s):

Galerkin Method ◽

Discontinuous Galerkin ◽

Discontinuous Galerkin Method ◽

Space Time ◽

Time Dependent ◽

Advection Diffusion ◽

Diffusion Problems

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The Space–Time Kernel-Based Numerical Method for Burgers’ Equations

Mathematics ◽

10.3390/math6100212 ◽

2018 ◽

Vol 6 (10) ◽

pp. 212 ◽

Cited By ~ 3

Author(s):

Marjan Uddin ◽

Hazrat Ali

Keyword(s):

Numerical Scheme ◽

Linear Equations ◽

Time Integration ◽

Time Dependent ◽

Burgers Equations ◽

Time Stepping ◽

Temporal Features ◽

Construct Differentiation ◽

Spatial Approximation ◽

Anisotropic Kernels

It is well known that major error occur in the time integration instead of the spatial approximation. In this work, anisotropic kernels are used for temporal as well as spatial approximation to construct a numerical scheme for solving nonlinear Burgers’ equations. The time-dependent PDEs are collocated in both space and time first, contrary to spatial discretization, and time stepping procedures for time integration are then applied. Physically one cannot in general expect that the spatial and temporal features of the solution behaves on the same order. Hence, one should have to incorporate anisotropic kernels. The nonlinear Burgers’ equations are converted by nonlinear transformation to linear equations. The spatial discretizations are carried out to construct differentiation matrices. Comparisons with most available numerical methods are made to solve the Burgers’ equations.

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A multi-timestep Robin–Robin domain decomposition method for time dependent advection-diffusion problems

Applied Mathematics and Computation ◽

10.1016/j.amc.2019.124596 ◽

2019 ◽

Vol 363 ◽

pp. 124596

Author(s):

Claudio Canuto ◽

Andrea Lo Giudice

Keyword(s):

Domain Decomposition ◽

Decomposition Method ◽

Domain Decomposition Method ◽

Time Dependent ◽

Advection Diffusion ◽

Diffusion Problems

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First-Order Hyperbolic System Method for Time-Dependent Advection-Diffusion Problems

10.21236/ada606728 ◽

2014 ◽

Cited By ~ 5

Author(s):

Alireza Mazaheri ◽

Hiroaki Nishikawa

Keyword(s):

Hyperbolic System ◽

Time Dependent ◽

First Order ◽

Advection Diffusion ◽

System Method ◽

Diffusion Problems

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Very efficient high-order hyperbolic schemes for time-dependent advection–diffusion problems: Third-, fourth-, and sixth-order

Computers & Fluids ◽

10.1016/j.compfluid.2014.06.020 ◽

2014 ◽

Vol 102 ◽

pp. 131-147 ◽

Cited By ~ 16

Author(s):

Alireza Mazaheri ◽

Hiroaki Nishikawa

Keyword(s):

High Order ◽

Sixth Order ◽

Time Dependent ◽

Advection Diffusion ◽

Diffusion Problems

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On an Improved Time Integration Scheme for Stiff Constitutive Models of Inelastic Deformation

Journal of Engineering Materials and Technology ◽

10.1115/1.3225820 ◽

1985 ◽

Vol 107 (4) ◽

pp. 282-285 ◽

Cited By ~ 15

Author(s):

Vinod Banthia ◽

Subrata Mukherjee

Keyword(s):

Strain Rate ◽

Inelastic Deformation ◽

Time Integration ◽

Constitutive Models ◽

Size Control ◽

Integration Scheme ◽

Step Size ◽

Time Step ◽

Time Step Size ◽

Step Size Control

For the time-integration of stiff constitutive models of inelastic deformation, the explicit (one step Euler) integration scheme can be used provided the time step size is closely monitored and controlled. The time step size control scheme based on prescribed error bounds is of limited use because it requires an a priori estimate of the maximum nonelastic strain rate for the selection of a proper error bound. In this paper, a new scheme for time-step size control is presented. This scheme automatically scales the time-step size by the maximum nonelastic strain rate. That the new scheme is superior to the old one is evident from the results of the various problems presented here.

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