Article
Computer Science, Interdisciplinary Applications
Brian C. Vermeire, Siavash Hedayati Nasab
Summary: This paper introduces a family of accelerated implicit-explicit (AIMEX) schemes for solving stiff systems of equations. AIMEX schemes can significantly improve stability and allowable time step sizes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Brian C. Vermeire
Summary: Recently, Paired Explicit Runge-Kutta (P-ERK) schemes were introduced to accelerate the solution of locally-stiff systems of equations by significantly reducing computational cost. In this work, a framework for incorporating an embedded pair within the original P-ERK formulation is presented. Numerical results demonstrate that adaptive time stepping using embedded P-ERK schemes yields excellent agreement with reference data while being up to seven times less computationally expensive than classical embedded pairs for all cases.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Siavash Hedayati Nasab, Brian C. Vermeire
Summary: This paper introduces a new family of Paired Explicit Runge-Kutta (P-ERK) methods that can be chosen for different Runge-Kutta schemes based on local stiffness criteria. By optimizing these schemes and verifying their effectiveness in nonlinear systems of equations, experimental results show that this new family of third-order P-ERK schemes performs accurately and efficiently in solving locally-stiff systems of equations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Severiano Gonzalez-Pinto, Domingo Hernandez-Abreu, Maria S. Perez-Rodriguez, Arash Sarshar, Steven Roberts, Adrian Sandu
Summary: This work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge-Kutta (GARK) methods. New IMIM-GARK splitting methods are developed and tested using parabolic systems. Classical splitting methods can be studied in this unified framework.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Emil M. Constantinescu
Summary: We propose a new method that extends conservative explicit multirate methods to implicit explicit-multirate methods. We develop extensions of order one and two with different stability properties on the implicit side. The method is suitable for time-stepping adaptive mesh refinement PDE discretizations with different degrees of stiffness. A numerical example with an advection-diffusion problem illustrates the new method's properties.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Manuel Calvo, Lin Fu, Juan I. Montijano, Luis Randez
Summary: This paper proposes new explicit integrators for numerical solution of stiff evolution equations. The new integrators have simpler computation process and better stability properties compared to previous methods. A series of numerical experiments demonstrates that the new integrators provide a simple and stable solver for stiff systems.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Assyr Abdulle, Giacomo Rosilho de Souza
Summary: Stabilized explicit methods are efficient for large systems of stiff stochastic differential equations (SDEs), but lose efficiency when severe stiffness is induced by a few fast degrees of freedom. Therefore, by introducing a stochastic modified equation whose stiffness depends solely on the slow terms and integrating it with a stabilized explicit scheme, a multirate method is devised to overcome the bottleneck caused by severely stiff terms and recover the efficiency of stabilized schemes in large systems of nonlinear SDEs.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Ben S. Southworth, Oliver Krzysik, Will Pazner, Hans De Sterck
Summary: This paper introduces a theoretical and algorithmic preconditioning framework for solving the systems of equations that arise from fully implicit Runge-Kutta methods applied to linear numerical PDEs. The preconditioned operator is proven to have a condition number bounded by a small constant, independent of the spatial mesh and time-step size, and with weak dependence on number of stages/polynomial order.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Davide Torlo, Philipp Oeffner, Hendrik Ranocha
Summary: This article discusses the methods to analyze the performance and robustness of Patankar-type schemes, and demonstrates their problematic behavior on both linear and nonlinear stiff problems.
APPLIED NUMERICAL MATHEMATICS
(2022)
Review
Engineering, Aerospace
Yongle Du, John A. Ekaterinaris
Summary: This review discusses the maturity of computational fluid dynamics methods and their benefits in fluid dynamics research and engineering applications. It highlights the advantages of high order numerical schemes, stability and error analysis, as well as the challenges and benefits of implicit time advancement. The review also covers recent developments in enhanced stability and error control, solution techniques for parallel computing, and the enforcement of implicit boundary conditions.
PROGRESS IN AEROSPACE SCIENCES
(2022)
Article
Engineering, Mechanical
Linhe Ge, Yang Zhao, Shouren Zhong, Zitong Shan, Fangwu Ma, Zhiwu Han, Konghui Guo
Summary: This study explores the application of nonlinear model predictive control in autonomous driving and proposes a method to solve the integration stability problem. By using the stabilized explicit Runge-Kutta integration method and integrating it into the gradient-based model predictive control framework, the problem of computational stability at low speeds is successfully addressed, improving the computational efficiency and eliminating steady-state errors.
NONLINEAR DYNAMICS
(2023)
Article
Mathematics, Applied
Assyr Abdulle, Marcus J. Grote, Giacomo Rosilho De Souza
Summary: Stabilized Runge-Kutta methods are efficient for solving large systems of stiff nonlinear differential equations, but their efficiency deteriorates when only a few components induce stiffness. We propose a modified equation and an explicit multirate Runge-KuttaChebyshev (mRKC) method with stability conditions independent of severely stiff components, which is demonstrated to be stable and efficient through numerical experiments.
MATHEMATICS OF COMPUTATION
(2022)
Article
Mathematics, Applied
Yayun Fu, Zhuangzhi Xu
Summary: In this paper, we propose an explicit energy-preserving Runge-Kutta scheme for solving the nonlinear Schrodinger equation. By introducing an auxiliary variable and using the projection technique, we construct a stable discrete scheme and provide stability results. Extensive numerical examples demonstrate the high accuracy, energy preservation, and effectiveness of the proposed scheme in long time simulation.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Md Masud Rana, Victoria E. Howle, Katharine Long, Ashley Meek, William Milestone
Summary: A new preconditioner based on LDU factorization and algebraic multigrid subsolves is introduced for scalability in implicit Runge-Kutta time integration of large, structured systems. Experimental results show that this new preconditioner outperforms others, especially as spatial discretization becomes more refined and temporal order is increased.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Chuanjun Chen, Tong Zhang
Summary: In this paper, the stability of three implicit/explicit (IMEX) schemes for the time-dependent natural convection problem is analyzed. These schemes include the first order backward Euler scheme, second order Crank-Nicolson IMEX scheme and BDF2-AB2 combination. The linear terms are treated implicitly and the nonlinear terms explicitly in all schemes. By splitting the original nonlinear problem into two linearized subproblems with constant coefficient matrixes, the computational complexity is reduced and storage is saved. The main contribution of this paper is establishing the unconditional stability of three IMEX schemes for incompressible flows, which improves and supplements existing theoretical findings. Numerical examples are provided to demonstrate the performance of the considered numerical schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Erik Burman, Rebecca Durst, Johnny Guzman
Summary: The study analyzes a splitting method for a canonical fluid structure interaction problem, which uses a Robin-Robin boundary condition to define an explicit coupling between the fluid and the structure. The method is proven to be stable, and an error estimate is provided showing the error at the final time T is O(T Delta t), where Delta t is the time step.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Engineering, Multidisciplinary
Rodrigo C. Moura, Andrea Cassinelli, Andre F. C. da Silva, Erik Burman, Spencer J. Sherwin
Summary: One of the strengths of the discontinuous Galerkin method is its balance between accuracy and robustness, which is particularly useful in high Reynolds-number flow simulations. This paper introduces a new gradient jump penalisation (GJP) approach that shows potential equivalent to DG dissipation and superior to previous SVV approaches. The GJP stabilisation approach is supported by eigenanalysis and turbulent flow simulations, achieving higher quality flow solutions compared to SVV.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Engineering, Multidisciplinary
Erik Burman, Omar Duran, Alexandre Ern
Summary: The study proposes an unfitted high-order method for solving the wave equation in domains with different material properties. The method demonstrates optimal convergence rates for smooth solutions and accurately simulates the propagation of acoustic waves in heterogeneous media.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Erik Burman, Johnny Guzman
Summary: This paper investigates a finite element method with symmetric stabilisation for solving the transient convection-diffusion equation. The convection term and stabilisation are explicitly treated using an extrapolated approximate solution. The stability and error estimates of the method are proven under specific conditions and illustrated with numerical examples.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Erik Burman, Peter Hansbo, Mats G. Larson
Summary: In this paper, we study the effect of the Smagorinsky model on the stability of the associated perturbation equation. We demonstrate that in the presence of a spectral gap, the Smagorinsky model provides stability estimates for perturbations with exponential growth, depending only on the large scale gradient. Furthermore, we show that in the context of stabilized finite element methods, the Smagorinsky model acts as a stabilizer and yields close to optimal error estimates for smooth flows in the pre-asymptotic high Reynolds number regime.
JOURNAL OF MATHEMATICAL FLUID MECHANICS
(2022)
Article
Mathematics, Applied
Erik Burman, Riccardo Puppi
Summary: This study introduces two different discrete formulations for handling the weak imposition of Neumann boundary conditions in Darcy flow, using mixed finite elements on different types of meshes to achieve optimal convergent numerical schemes. Both methods are proven to be stable and capable of obtaining optimal error estimates for the velocity field.
JOURNAL OF NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Timo Betcke, Erik Burman, Matthew W. Scroggs
Summary: This paper considers the use of boundary element methods to solve boundary condition problems, where the Calderón projector is used as the system matrix and boundary conditions are weakly imposed using a variational boundary operator designed with augmented Lagrangian methods. Both the primal trace variable and the flux are approximated regardless of the boundary conditions. The paper specifically focuses on the imposition of Dirichlet conditions on the Helmholtz equation and extends the analysis of the Laplace problem to this case. The theory is demonstrated through numerical examples.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics
Erik Burman, Miguel A. Fernandez, Fannie M. Gerosa
Summary: In this paper, an error analysis of the semi-implicit scheme is performed for the linear fluid-structure interaction system. The analysis shows that the leading term in the energy error scales as O(h(r-1/2)) under a hyperbolic-CFL condition, where r=1,2 represents the extrapolation order of the solid velocity. Numerical experiments illustrate the advantages of the considered method over standard loosely coupled schemes and its comparable accuracy to the strongly coupled scheme.
VIETNAM JOURNAL OF MATHEMATICS
(2023)
Review
Computer Science, Interdisciplinary Applications
Erik Burman, Peter Hansbo, Mats G. Larson
Summary: In this paper, recent advances in the application of the augmented Lagrange multiplier method as a general approach for generating multiplier-free stabilised methods are reviewed. The method combines a standard Lagrange multiplier method with a penalty term to penalise the constraint equations, and is commonly used in iterative algorithms for constrained optimisation problems. The paper first explains how the method can generate Galerkin/Least Squares type schemes for equality constraints, and then extends it to develop new stabilised methods for inequality constraints. Several examples of its application to different problems in computational mechanics are presented.
ARCHIVES OF COMPUTATIONAL METHODS IN ENGINEERING
(2023)
Article
Engineering, Multidisciplinary
Erik Burman, Peter Hansbo, Mats G. Larson, Karl Larsson
Summary: In this article, we develop a discrete extension operator for trimmed spline spaces, which consist of piecewise polynomial functions with k continuous derivatives. The construction of the operator relies on polynomial extension from neighboring elements and projection back into the spline space. We prove stability and approximation results for the extension operator, and demonstrate how it can be used to construct a stable cut isogeometric method for solving an elliptic model problem.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Erik Burman, Jurriaan J. J. Gillissen, Lauri Oksanen
Summary: In this paper, we analyze the stability of the system of partial differential equations used in scalar image velocimetry. We revisit a numerical technique for reconstructing velocity vectors from images of a passive scalar field and investigate its stability in synthetic scalar fields generated by numerical simulation. Furthermore, we present a mathematical analysis of the nonlinear coupled problem and prove that, in the two-dimensional case, the solutions of the Navier-Stokes equations can be uniquely determined by the measured scalar field. A conditional stability estimate is also provided.
JOURNAL OF INVERSE AND ILL-POSED PROBLEMS
(2023)
Article
Mathematics, Interdisciplinary Applications
Erik Burman, Janosch Preuss
Summary: We present a stabilized finite element method for solving the unique continuation problem of the time-harmonic elastic wave equation with variable coefficients. By using conditional stability estimates, we prove convergence rates of the proposed method which account for the noise level and polynomial degree. Numerical experiments confirm our theoretical results and investigate the influence of domain geometry on the reconstruction quality. We find that certain convexity properties are essential for accurate recovery of the wave displacement outside the data domain, and higher polynomial orders can be more efficient but also more sensitive to the ill-conditioned nature of the problem.
GEM-INTERNATIONAL JOURNAL ON GEOMATHEMATICS
(2023)
Article
Engineering, Multidisciplinary
Erik Burman, Peter Hansbo, Mats G. Larson, Karl Larsson
Summary: This article explores the application of Isogeometric analysis, Galerkin Least-Squares methods, and Augmented Lagrangian techniques to solve nonstandard problems. It particularly focuses on continuation problems involving second-order partial differential equations with incomplete boundary data and measurements of the solution on a subdomain. The use of higher regularity spline spaces leads to simplified formulations and potential minimal multiplier space, ensuring inf-sup stability and optimal order convergence given appropriate a priori assumptions.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Erik Burman, Rebecca Durst, Miguel Fernandez, Johnny Guzman
Summary: We propose a new loosely coupled, non-iterative time-splitting scheme based on Robin-Robin coupling conditions. We provide a unified analysis for this scheme applied to both parabolic/parabolic and parabolic/hyperbolic coupled systems. We demonstrate that the scheme is stable and the error convergence rate is Phi(?delta t root?T + log(1/delta t))? where delta t is the time step.
JOURNAL OF NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Erik Burman, Omar Duran, Alexandre Ern
Summary: This article presents hybrid high-order methods for the acoustic wave equation in the time domain, including second-order and first-order system formulations with different numerical discretization schemes. Numerical results demonstrate that these methods exhibit optimal convergence rates for smooth solutions and wave propagation scenarios.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2022)
