Convergence rates of the front tracking method for conservation laws in the Wasserstein distances

We prove that front tracking approximations to entropy solutions of scalar conservation laws with convex fluxes converge at a rate of $\Delta x^2$ in the 1-Wasserstein distance $W_1$. Assuming positive initial data, we also show that the approximations converge at a rate of $\Delta x$ in the $\infty$-Wasserstein distance $W_\infty$. Moreover, from a simple interpolation inequality between $W_1$ and $W_\infty$ we obtain convergence rates in all the $p$-Wasserstein distances: $\Delta x^{1+1/p}$, $p \in [1,\infty]$.Comment: Improved the introduction. Added lemma 4.1 and did some smaller change

Numerical conservative solutions of the Hunter--Saxton equation

In the article a convergent numerical method for conservative solutions of the Hunter--Saxton equation is derived. The method is based on piecewise linear projections, followed by evolution along characteristics where the time step is chosen in order to prevent wave breaking. Convergence is obtained when the time step is proportional to the square root of the spatial step size, which is a milder restriction than the common CFL condition for conservation laws

The optimal convergence rate of monotone schemes for conservation laws in the Wasserstein distance

In 1994, Nessyahu, Tadmor and Tassa studied convergence rates of monotone finite volume approximations of conservation laws. For compactly supported, \Lip^+-bounded initial data they showed a first-order convergence rate in the Wasserstein distance. Our main result is to prove that this rate is optimal. We further provide numerical evidence indicating that the rate in the case of \Lip^+-unbounded initial data is worse than first-order.Comment: 10 pages, 5 figures, 2 tables. Fixed typos. Article published in Journal of Scientific Computin

On complex dynamics in a Purkinje and a ventricular cardiac cell model

Cardiac muscle cells can exhibit complex patterns including irregular behaviour such as chaos or (chaotic) early afterdepolarisations (EADs), which can lead to sudden cardiac death. Suitable mathematical models and their analysis help to predict the occurrence of such phenomena and to decode their mechanisms. The focus of this paper is the investigation of dynamics of cardiac muscle cells described by systems of ordinary differential equations. This is generically performed by studying a Purkinje cell model and a modified ventricular cell model. We find chaotic dynamics with respect to the leak current in the Purkinje cell model, and EADs and chaos with respect to a reduced fast potassium current and an enhanced calcium current in the ventricular cell model -- features that have been experimentally observed and are known to exist in some models, but are new to the models under present consideration. We also investigate the related monodomain models of both systems to study synchronisation and the behaviour of the cells on macro-scale in connection with the discovered features. The models show qualitatively the same behaviour to what has been experimentally observed. However, for certain parameter settings the dynamics occur within a non-physiological range

On complex dynamics in a Purkinje and a ventricular cardiac cell model

Cardiac muscle cells can exhibit complex patterns including irregular behaviour such as chaos or (chaotic) early afterdepolarisations (EADs), which can lead to sudden cardiac death. Suitable mathematical models and their analysis help to predict the occurrence of such phenomena and to decode their mechanisms. The focus of this paper is the investigation of dynamics of cardiac muscle cells described by systems of ordinary differential equations. This is generically performed by studying a Purkinje cell model and a modified ventricular cell model. We find chaotic dynamics with respect to the leak current in the Purkinje cell model, and EADs and chaos with respect to a reduced fast potassium current and an enhanced calcium current in the ventricular cell model — features that have been experimentally observed and are known to exist in some models, but are new to the models under present consideration. We also investigate the related monodomain models of both systems to study synchronisation and the behaviour of the cells on macro-scale in connection with the discovered features. The models show qualitatively the same behaviour to what has been experimentally observed. However, for certain parameter settings the dynamics occur within a non-physiological range

Noise-driven bifurcations in a neural field system modelling networks of grid cells

The activity generated by an ensemble of neurons is affected by various noise sources. It is a well-recognised challenge to understand the effects of noise on the stability of such networks. We demonstrate that the patterns of activity generated by networks of grid cells emerge from the instability of homogeneous activity for small levels of noise. This is carried out by analysing the robustness of network activity patterns with respect to noise in an upscaled noisy grid cell model in the form of a system of partial differential equations. Inhomogeneous network patterns are numerically understood as branches bifurcating from unstable homogeneous states for small noise levels. We show that there is a phase transition occurring as the level of noise decreases. Our numerical study also indicates the presence of hysteresis phenomena close to the precise critical noise value.publishedVersio

Relaxation Systems with Applications to Two-Phase Flow

Relaxation systems are widely studied and much used to describe nonequilibrium phenomena, occurring in, for example, two-phase flow. In this thesis we will therefore consider relaxation systems in one space dimension and get some insight into their applications to two-phase flow. Two main topics will be considered: hyperbolic constant-coefficient relaxation systems and a specific two-phase model. Entropy conditions are also studied. All three topics are connected with relaxation processes.The first part consists of a study of the transitional wave-dynamics of strictly hyperbolic constant-coefficient relaxation systems with stable rank one relaxation matrices. By realizing that the eigenvalue polynomial of such a system can be written as a convex sum of the two eigenvalue polynomials of the corresponding formal limiting systems, we show that the system is stable if and only if it fulfills the interlacing property known as the subcharacteristic condition. Further, if the system is stable, it is shown that the transitional wave-velocities can never exceed the velocities of the corresponding homogeneous system. The results are applied to a two-phase model. Mathematical entropy is studied in connection with conservative and nonconservative relaxation systems. Beneficial properties, such as symmetry and the fulfillment of the subcharacteristic condition, follow directly from the existence of a convex entropy for conservative systems. This does not hold in general for nonconservative systems. A two-phase model with a well-reservoir interaction term and a viscous term is studied. The estimates that exist for the full model are relaxation parameter dependent. An existence result for the reduced model, the formal limit model as the relaxation time tends to $0$, does therefore not follow directly from the existence result for the full model. A new existence result for the reduced model is therefore achieved in a similar way to that of the full model. It relies on the assumption that specific parameters and initial conditions are small enough. The result also ensures that both phases will exist at any spatial point for any finite time

Relaxation Systems with Applications to Two-Phase Flow

Relaxation systems are widely studied and much used to describe nonequilibrium phenomena, occurring in, for example, two-phase flow. In this thesis we will therefore consider relaxation systems in one space dimension and get some insight into their applications to two-phase flow. Two main topics will be considered: hyperbolic constant-coefficient relaxation systems and a specific two-phase model. Entropy conditions are also studied. All three topics are connected with relaxation processes.The first part consists of a study of the transitional wave-dynamics of strictly hyperbolic constant-coefficient relaxation systems with stable rank one relaxation matrices. By realizing that the eigenvalue polynomial of such a system can be written as a convex sum of the two eigenvalue polynomials of the corresponding formal limiting systems, we show that the system is stable if and only if it fulfills the interlacing property known as the subcharacteristic condition. Further, if the system is stable, it is shown that the transitional wave-velocities can never exceed the velocities of the corresponding homogeneous system. The results are applied to a two-phase model. Mathematical entropy is studied in connection with conservative and nonconservative relaxation systems. Beneficial properties, such as symmetry and the fulfillment of the subcharacteristic condition, follow directly from the existence of a convex entropy for conservative systems. This does not hold in general for nonconservative systems. A two-phase model with a well-reservoir interaction term and a viscous term is studied. The estimates that exist for the full model are relaxation parameter dependent. An existence result for the reduced model, the formal limit model as the relaxation time tends to $0$, does therefore not follow directly from the existence result for the full model. A new existence result for the reduced model is therefore achieved in a similar way to that of the full model. It relies on the assumption that specific parameters and initial conditions are small enough. The result also ensures that both phases will exist at any spatial point for any finite time

Analysis and simulation of a modified cardiac cell model gives accurate predictions of the dynamics of the original one

The 19-dimensional TP06 cardiac muscle cell model is reduced to a 17-dimensional version, which satisfies the required conditions for performing an analysis of its dynamics by means of bifurcation theory. The reformulated model is shown to be a good approximation of the original one. As a consequence, one can extract fairly precise predictions of the behaviour of the original model from the bifurcation analysis of the modified model. Thus, the findings of bifurcations linked to complex dynamics in the modified model - like early afterdepolarisations (EADs), which can be precursors to cardiac death - predicts the occurrence of the same dynamics in the original model. It is shown that bifurcations linked to EADs in the modified model accurately predicts EADs in the original model at the single cell level. Finally, these bifurcations are linked to wave break-up leading to cardiac death at the tissue level