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Register a new userSecond order splitting of a class of fourth order PDEs with point constraints
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Added by
Philip Herbert
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential equations with point constraints using the idea of splitting into coupled second order equations. An approach is formulated using a penalty method to impose the constraints. Our main motivation is to treat certain fourth order equations involving the biharmonic operator and point Dirichlet constraints for example arising in the modelling of biomembranes on curved and flat surfaces but the approach may be applied more generally. The theory for well-posedness and approximation is presented in an abstract setting. Several examples are described together with some numerical experiments.
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We formulate a well-posedness and approximation theory for a class of generalised saddle point problems. In this way we develop an approach to a class of fourth order elliptic partial differential equations using the idea of splitting into coupled second order equations. Our main motivation is to treat certain fourth order surface equations arising in the modelling of biomembranes but the approach may be applied more generally. In particular, we are interested in equations with non-smooth right hand sides and operators which have non-trivial kernels.The theory for well posedness and approximation is presented in an abstract setting. Several examples are described together with some numerical experiments.
For a class of fourth order gradient flow problems, integration of the scalar auxiliary variable (SAV) time discretization with the penalty-free discontinuous Galerkin (DG) spatial discretization leads to SAV-DG schemes. These schemes are linear and shown unconditionally energy stable. But the reduced linear systems are rather expensive to solve due to the dense coefficient matrices. In this paper, we provide a procedure to pre-evaluate the auxiliary variable in the piecewise polynomial space. As a result, the computational complexity of $O(mathcal{N}^2)$ reduces to $O(mathcal{N})$ when exploiting the conjugate gradient (CG) solver. This hybrid SAV-DG method is more efficient and able to deliver satisfactory results of high accuracy. This was also compared with solving the full augmented system of the SAV-DG schemes.
A second-order accurate in time, positivity-preserving, and unconditionally energy stable operator splitting numerical scheme is proposed and analyzed for the system of reaction-diffusion equations with detailed balance. The scheme is designed based on an energetic variational formulation, in which the reaction part is reformulated in terms of the reaction trajectory, and both the reaction and diffusion parts dissipate the same free energy. At the reaction stage, the reaction trajectory equation is approximated by a second-order Crank-Nicolson type method. The unique solvability, positivity-preserving, and energy-stability are established based on a convexity analysis. In the diffusion stage, an exact integrator is applied if the diffusion coefficients are constant, and a Crank-Nicolson type scheme is applied if the diffusion process becomes nonlinear. In either case, both the positivity-preserving property and energy stability could be theoretically established. Moreover, a combination of the numerical algorithms at both stages by the Strang splitting approach leads to a second-order accurate, structure-preserving scheme for the original reaction-diffusion system. Numerical experiments are presented, which demonstrate the accuracy of the proposed scheme.
In this work, we determine the full expression of the local truncation error of hyperbolic partial differential equations (PDEs) on a uniform mesh. If we are employing a stable numerical scheme and the global solution error is of the same order of accuracy as the global truncation error, we make the following observations in the asymptotic regime, where the truncation error is dominated by the powers of $Delta x$ and $Delta t$ rather than their coefficients. Assuming that we reach the asymptotic regime before the machine precision error takes over, (a) the order of convergence of stable numerical solutions of hyperbolic PDEs at constant ratio of $Delta t$ to $Delta x$ is governed by the minimum of the orders of the spatial and temporal discretizations, and (b) convergence cannot even be guaranteed under only spatial or temporal refinement. We have tested our theory against numerical methods employing Method of Lines and not against ones that treat space and time together, and we have not taken into consideration the reduction in the spatial and temporal orders of accuracy resulting from slope-limiting monotonicity-preserving strategies commonly applied to finite volume methods. Otherwise, our theory applies to any hyperbolic PDE, be it linear or non-linear, and employing finite difference, finite volume, or finite element discretization in space, and advanced in time with a predictor-corrector, multistep, or a deferred correction method. If the PDE is reduced to an ordinary differential equation (ODE) by specifying the spatial gradients of the dependent variable and the coefficients and the source terms to be zero, then the standard local truncation error of the ODE is recovered. We perform the analysis with generic and specific hyperbolic PDEs using the symbolic algebra package SymPy, and conduct a number of numerical experiments to demonstrate our theoretical findings.
Discrete approximations to the equation begin{equation*} L_{cont}u = u^{(4)} + D(x) u^{(3)} + A(x) u^{(2)} + (A(x)+H(x)) u^{(1)} + B(x) u = f, ; xin[0,1] end{equation*} are considered. This is an extension of the Sturm-Liouville case $D(x)equiv H(x)equiv 0$ [ M. Ben-Artzi, J.-P. Croisille, D. Fishelov and R. Katzir, Discrete fourth-order Sturm-Liouville problems, IMA J. Numer. Anal. {bf 38} (2018), 1485-1522. doi: 10.1093/imanum/drx038] to the non-self-adjoint setting. The natural boundary conditions in the Sturm-Liouville case are the values of the function and its derivative. The inclusion of a third-order discrete derivative entails a revision of the underlying discrete functional calculus. This revision forces evaluations of accurate discrete approximations to the boundary values of the second, third and fourth order derivatives. The resulting functional calculus provides the discrete analogs of the fundamental Sobolev properties--compactness and coercivity. It allows to obtain a general convergence theorem of the discrete approximations to the exact solution. Some representative numerical examples are presented.
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