Two-step error estimators for implicit Runge--Kutta methods applied to stiff systems

González-Pinto, S.; Montijano, J. I.; Pérez-Rodríguez, S.

doi:10.1145/974781.974782

Cited by 13 publications

(14 citation statements)

References 9 publications

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“…where B is the mass matrix and X and U are the vectors representing the mesh and solution, respectively. This ODE system is integrated from t n to t n+1 = t n + ∆t n using the fifth-order Radau IIA method, with ∆t n being determined by a standard time step size selection procedure (e.g., see Hairer et al [23,Section II.4]) and using a two-step error estimator of Gonzalez-Pinto et al [22]. The relative and absolute tolerances rtol = 10 −6 and atol = 10 −8 are taken in the computation.…”

Section: Finite Element Discretizationmentioning

confidence: 99%

A study on moving mesh finite element solution of the porous medium equation

Ngo

Huang

2017

Journal of Computational Physics

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An adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption. The method is based on the moving mesh partial differential equation approach and employs its newly developed implementation. The implementation has several improvements over the traditional one, including its explicit, compact form of the mesh velocities, ease to program, and less likelihood of producing singular meshes. Three types of metric tensor that correspond to uniform and arclength-based and Hessian-based adaptive meshes are considered. The method shows first-order convergence for uniform and arclength-based adaptive meshes, and second-order convergence for Hessian-based adaptive meshes. It is also shown that the method can be used for situations with complex free boundaries, emerging and splitting of free boundaries, and the porous medium equation with variable exponents and absorption. Two-dimensional numerical results are presented.

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Section: Finite Element Discretizationmentioning

confidence: 99%

A study on moving mesh finite element solution of the porous medium equation

Ngo

Huang

2017

Journal of Computational Physics

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“…The fifth-order Radau IIA method is used in time discretization; if you are interested in more details, please refer to [44]. From the method, we can determine the time step̃; if the real time step̃< , the next time level, the mesh coordinate, and the time step of next time level update: +1 ← +̃, X +1 ← X +̃Ẋ , and +1 ←̃.…”

Section: The Implementation Of the Methods Combined With Finite Elemenmentioning

confidence: 99%

Finite Element Method of BBM-Burgers Equation with Dissipative Term Based on Adaptive Moving Mesh

Gao

et al. 2017

Discrete Dynamics in Nature and Society

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A finite element model is proposed for the Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation with a high-order dissipative term; the scheme is based on adaptive moving meshes. The model can be applied to the equations with spatial-time mixed derivatives and high-order derivative terms. In this scheme, new variables are needed to make the equation become a coupled system, and then the linear finite element method is used to discretize the spatial derivative and the fifth-order Radau IIA method is used to discretize the time derivative. The simulations of 1D and 2D BBM-Burgers equations with high-order dissipative terms are presented in numerical examples. The numerical results show that the method keeps a second-order convergence in space and provides a smaller error than that based on the fixed mesh, which demonstrates the effectiveness and feasibility of the finite element method based on the moving mesh. We also study the effect of the dissipative terms with different coefficients in the equation; by numerical simulations, we find that the dissipative term plays a more important role than in dissipation.

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“…Substituting , into and taking ψ = ϕ i , i = 1, …, N vi successively, we get

\begin{array}{l} \sum_{j = 1}^{N_{italicvi}} ((), \int_{Ω ((), t)} ϕ_{j} ϕ_{i} d bold-italicx) \frac{{italicdv}_{j}}{dt} = \int_{Ω ((), t)} \nabla v_{h} \cdot ((), \overset{X}{true} ϕ_{i} - {mv}_{h} \nabla ϕ_{i}) d bold-italicx \\ + ((), 1 - m) \int_{Ω ((), t)} {|\nabla v_{h}|}^{2} ϕ_{i} d bold-italicx, i = 1, \dots, N_{vi}, \end{array}

which can be written into the matrix form as

B (bold-italicX) \dot{bold-italicV} = F (bold-italicV, bold-italicX, \overset{X}{true}),

where B is the mass matrix, and X and V are vectors representing the mesh and solution, respectively. The ODE system is integrated from t n to t n +1 using the fifth‐order Radau IIA method (an implicit Runge–Kutta method) with a standard time step selection procedure where the relative and absolute tolerance are chosen as 10 −6 and 10 −8 , respectively, and the error estimation is based on a two‐step error estimator of Gonzalez‐Pinto et al .…”

Section: The Moving Mesh Femmentioning

confidence: 99%

Adaptive finite element solution of the porous medium equation in pressure formulation

Ngo

Huang

2019

Numerical Methods Partial

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The pressure formulation of the porous medium equation has been commonly used in theoretical studies due to its much better regularities than the original formulation. The goal here is to study its use in the adaptive moving mesh finite element solution. The free boundary is traced explicitly through Darcy's law. The method is shown numerically second‐order in space and first‐order in time in the pressure variable. Moreover, the convergence order of the error in the location of the free boundary is almost second‐order in the maximum norm. However, numerical results also show that the convergence order in the original variable stays between first‐order and second‐order in L1 norm or between 0.5th‐order and first‐order in L2 norm. Nevertheless, the current method can offer some advantages over numerical methods based on the original formulation for situations with large exponents or when a more accurate location of the free boundary is desired.

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Two-step error estimators for implicit Runge--Kutta methods applied to stiff systems

Cited by 13 publications

References 9 publications

A study on moving mesh finite element solution of the porous medium equation

A study on moving mesh finite element solution of the porous medium equation

Finite Element Method of BBM-Burgers Equation with Dissipative Term Based on Adaptive Moving Mesh

Adaptive finite element solution of the porous medium equation in pressure formulation

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