Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization

In this work, we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping. The method relies on an artificial diffusion method, based on a graph-Laplacian operator. It is nonlinear, since it depends on a shock detector. Further, the resulting method is linearity preserving. The same shock detector is used to gradually lump the mass matrix. The resulting method is LED, positivity preserving, and also satisfies a global DMP. Lipschitz continuity has also been proved. However, the resulting scheme is highly nonlinear, leading to very poor nonlinear convergence rates. We propose a smooth version of the scheme, which leads to twice differentiable nonlinear stabilization schemes. It allows one to straightforwardly use Newton’s method and obtain quadratic convergence. In the numerical experiments, steady and transient linear transport, and transient Burgers’ equation have been considered in 2D. Using the Newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of Anderson acceleration with the original non-smooth scheme. In any case, these properties are only true for the converged solution, but not for iterates. In this sense, we have also proposed the concept of projected nonlinear solvers, where a projection step is performed at the end of every nonlinear iterations onto a FE space of admissible solutions. The space of admissible solutions is the one that satisfies the desired monotonic properties (maximum principle or positivity).Peer ReviewedPostprint (author's final draft

Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization

In this work, we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping. The method relies on an artificial diffusion method, based on a graph-Laplacian operator. It is nonlinear, since it depends on a shock detector. Further, the resulting method is linearity preserving. The same shock detector is used to gradually lump the mass matrix. The resulting method is LED, positivity preserving, and also satisfies a global DMP. Lipschitz continuity has also been proved. However, the resulting scheme is highly nonlinear, leading to very poor nonlinear convergence rates. We propose a smooth version of the scheme, which leads to twice differentiable nonlinear stabilization schemes. It allows one to straightforwardly use Newton’s method and obtain quadratic convergence. In the numerical experiments, steady and transient linear transport, and transient Burgers’ equation have been considered in 2D. Using the Newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of Anderson acceleration with the original non-smooth scheme. In any case, these properties are only true for the converged solution, but not for iterates. In this sense, we have also proposed the concept of projected nonlinear solvers, where a projection step is performed at the end of every nonlinear iterations onto a FE space of admissible solutions. The space of admissible solutions is the one that satisfies the desired monotonic properties (maximum principle or positivity)

Maximum-principle preserving space-time isogeometric analysis

In this work we propose a nonlinear stabilization technique for convection-diffusion-reaction and pure transport problems discretized with space-time isogeometric analysis. The stabilization is based on a graph-theoretic artificial diffusion operator and a novel shock detector for isogeometric analysis. Stabilization in time and space directions are performed similarly, which allow us to use high-order discretizations in time without any CFL-like condition. The method is proven to yield solutions that satisfy the discrete maximum principle (DMP) unconditionally for arbitrary order. In addition, the stabilization is linearity preserving in a space-time sense. Moreover, the scheme is proven to be Lipschitz continuous ensuring that the nonlinear problem is well-posed. Solving large problems using a space-time discretization can become highly costly. Therefore, we also propose a partitioned space-time scheme that allows us to select the length of every time slab, and solve sequentially for every subdomain. As a result, the computational cost is reduced while the stability and convergence properties of the scheme remain unaltered. In addition, we propose a twice differentiable version of the stabilization scheme, which enjoys the same stability properties while the nonlinear convergence is significantly improved. Finally, the proposed schemes are assessed with numerical experiments. In particular, we considered steady and transient pure convection and convection-diffusion problems in one and two dimensions

Monotonicity-preserving finite element methods for hyperbolic problems

This thesis covers the development of monotonicity preserving finite element methods for hyperbolic problems. In particular, scalar convection-diffusion and Euler equations are used as model problems for the discussion in this dissertation. A novel artificial diffusion stabilization method has been proposed for scalar problems. This technique is proved to yield monotonic solutions, to be \ac{led}, Lipschitz continuous, and linearity preserving. These properties are satisfied in multiple dimensions and for general meshes. However, these results are limited to first order Lagrangian finite elements. A modification of this stabilization operator that is twice differentiable has been also proposed. With this regularized operator, nonlinear convergence is notably improved, while the stability properties remain unaltered (at least, in a weak sense). An extension of this stabilization method to high-order discretizations has also been proposed. In particular, arbitrary order space-time isogeometric analysis is used for this purpose. It has been proved that this scheme yields solutions that satisfy a global space-time discrete maximum principle unconditionally. A partitioned approach has also been proposed. This strategy reduces the computational cost of the scheme, while it preserves all stability properties. A regularization of this stabilization operator has also been developed. As for the first order finite element method, it improves the nonlinear convergence without harming the stability properties. An extension to Euler equations has also been pursued. In this case, instead of monotonicity-preserving, the developed scheme is local bounds preserving. Following the previous works, a regularized differentiable version has also been proposed. In addition, a continuation method using the parameters introduced for the regularization has been used. In this case, not only the nonlinear convergence is improved, but also the robustness of the method. However, the improvement in nonlinear convergence is limited to moderate tolerances and it is not as notable as for the scalar problem. Finally, the stabilized schemes proposed had been adapted to adaptive mesh refinement discretizations. In particular, nonconforming hierarchical octree-based meshes have been used. Using these settings, the efficiency of solving a monotonicity-preserving high-order stiff nonlinear problem has been assessed. Given a specific accuracy, the computational time required for solving the high-order problem is compared to the one required for solving a low-order problem (easy to converge) in a much finer adapted mesh. In addition, an error estimator based on the stabilization terms has been proposed and tested. The performance of all proposed schemes has been assessed using several numerical tests and solving various benchmark problems. The obtained results have been commented and included in the dissertation.La present tesi tracta sobre mètodes d'elements finits que preserven la monotonia per a problemes hiperbòlics. Concretament, els problemes que s'han utilitzat com a model en el desenvolupament d'aquesta tesi són l'equació escalar de convecció-difusió-reacció i les equacions d'Euler. Per a problemes escalars s'ha proposat un nou mètode d'estabilització mitjançant difusió artificial. S'ha provat que amb aquesta tècnica les solucions obtingudes són monòtones, l'esquema "disminueix els extrems locals", i preserva la linearitat. Aquestes propietats s'han pogut demostrar per múltiples dimensions i per malles generals. Per contra, aquests resultats només són vàlids per elements finits Lagrangians de primer ordre. També s'ha proposat una modificació de l'operador d'estabilització per tal de que aquest sigui diferenciable. Aquesta regularització ha permès millorar la convergència no-lineal notablement, mentre que les propietats d'estabilització no s'han vist alterades. L'anterior mètode d'estabilització s'ha adaptat a discretitzacions d'alt ordre. Concretament, s'ha utilitzat anàlisi isogeomètrica en espai i temps per a aquesta tasca. S'ha provat que les solucions obtingudes mitjançant aquest mètode satisfan el principi del màxim discret de forma global. També s'ha proposat un esquema particionat. Aquesta alternativa redueix el cost computacional, mentre preserva totes les propietats d'estabilitat. En aquest cas, també s'ha realitzat una regularització de l'operador d'estabilització per tal de que sigui diferenciable. Tal i com s'ha observat en els mètodes de primer ordre, aquesta regularització permet millorar la convergència no-lineal sense perdre les propietats d'estabilització. Posteriorment, s'ha estudiat l'adaptació dels mètodes anteriors a les equacions d'Euler. En aquest cas, en comptes de preservar la monotonia, l'esquema preserva "cotes locals". Seguint els desenvolupaments anteriors, s'ha proposat una versió diferenciable de l'estabilització. En aquest cas, també s'ha desenvolupat un mètode de continuació utilitzant els paràmetres introduïts per a la regularització. En aquest cas, no només ha millorat la convergència no-lineal sinó que l'esquema també esdevé més robust. Per contra, la millora en la convergència no-lineal només s'observa per a toleràncies moderades i no és tan notable com en el cas dels problemes escalars. Finalment, els esquemes d'estabilització proposat s'han adaptat a malles de refinament adaptatiu. Concretament, s'han utilitzat malles no-conformes basades en octrees. Utilitzant aquesta configuració, l'eficiència de resoldre un problema altament no-lineal ha estat avaluada de la següent forma. Donada una precisió determinada, el temps computacional requerit per resoldre el problema utilitzant un esquema d'alt ordre ha estat comparat amb el temps necessari per resoldre'l utilitzant un esquema de baix ordre en una malla adaptativa molt més refinada. Addicionalment, també s'ha proposat un estimador de l'error basat en l'operador d'estabilització. El comportament de tots els esquemes proposats anteriorment s'ha avaluat mitjançant varis tests numèrics. Els resultats s'han compilat i comentat en la present tesi.Postprint (published version

Differentiable monotonicity-preserving schemes for discontinuous Galerkin methods on arbitrary meshes

This work is devoted to the design of interior penalty discontinuous Galerkin (dG) schemes that preserve maximum principles at the discrete level for the steady transport and convection–diffusion problems and the respective transient problems with implicit time integration. Monotonic schemes that combine explicit time stepping with dG space discretization are very common, but the design of such schemes for implicit time stepping is rare, and it had only been attained so far for 1D problems. The proposed scheme is based on a piecewise linear dG discretization supplemented with an artificial diffusion that linearly depends on a shock detector that identifies the troublesome areas. In order to define the new shock detector, we have introduced the concept of discrete local extrema. The diffusion operator is a graph-Laplacian, instead of the more common finite element discretization of the Laplacian operator, which is essential to keep monotonicity on general meshes and in multi-dimension. The resulting nonlinear stabilization is non-smooth and nonlinear solvers can fail to converge. As a result, we propose a smoothed (twice differentiable) version of the nonlinear stabilization, which allows us to use Newton with line search nonlinear solvers and dramatically improve nonlinear convergence. A theoretical numerical analysis of the proposed schemes shows that they satisfy the desired monotonicity properties. Further, the resulting operator is Lipschitz continuous and there exists at least one solution of the discrete problem, even in the non-smooth version. We provide a set of numerical results to support our findings

A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws

We introduce a new methodology for adding localized, space-time smooth, artificial viscosity to nonlinear systems of conservation laws which propagate shock waves, rarefactions, and contact discontinuities, which we call the $C$-method. We shall focus our attention on the compressible Euler equations in one space dimension. The novel feature of our approach involves the coupling of a linear scalar reaction-diffusion equation to our system of conservation laws, whose solution $C(x,t)$ is the coefficient to an additional (and artificial) term added to the flux, which determines the location, localization, and strength of the artificial viscosity. Near shock discontinuities, $C(x,t)$ is large and localized, and transitions smoothly in space-time to zero away from discontinuities. Our approach is a provably convergent, spacetime-regularized variant of the original idea of Richtmeyer and Von Neumann, and is provided at the level of the PDE, thus allowing a host of numerical discretization schemes to be employed. We demonstrate the effectiveness of the $C$-method with three different numerical implementations and apply these to a collection of classical problems: the Sod shock-tube, the Osher-Shu shock-tube, the Woodward-Colella blast wave and the Leblanc shock-tube. First, we use a classical continuous finite-element implementation using second-order discretization in both space and time, FEM-C. Second, we use a simplified WENO scheme within our $C$-method framework, WENO-C. Third, we use WENO with the Lax-Friedrichs flux together with the $C$-equation, and call this WENO-LF-C. All three schemes yield higher-order discretization strategies, which provide sharp shock resolution with minimal overshoot and noise, and compare well with higher-order WENO schemes that employ approximate Riemann solvers, outperforming them for the difficult Leblanc shock tube experiment.Comment: 34 pages, 27 figure