The Numerical Solution of Differential Equations While Preserving the Geometric or Analytical Structure of The Dynamical System

The Numerical Solution of Differential Equations While Preserving the Geometric or Analytical Structure of The Dynamical System

Structure of the Literature Review

Framework

Introduction

A differential equation is a mathematical function that relates a function to one or more of its derivatives. It represents the variation or rates of change of select variables within the equation as it changes dynamically over time. There is a broad scope for the definition of a differential equation and the most relevant numerical conceptualization used here. In deriving numerical solutions to various problems, we may have two unknown numbers; firstly, we have an unknown as a single real number, or secondly, the unknown is a sequence of numbers that satisfy a given predetermined condition. A differential equation in this context is whereby an entire function is the unknown and the equation is what relates the function to its derivatives (Atkinson and Stewart, 2011).

Dynamic systems are systems where the defining equation contains one or more variables that change according to some given law or characteristic trend. An example is a time-varying system where the state of the system at any given point is a function of time and the outputs with the solutions vary for each setpoint. A more practical example of a dynamic system is a simple ideal pendulum. The motion and speed of the pendulum is a function of the forces acting on the pendulum concerning time. Even more straightforward practical visualization is the use of an object moving from a given point to another, the change in speed and distance of the object in motion is a function of time. The example above is a defined differential equation whereby the speed of the object from starting to the endpoint is a function of the difference in distance at the endpoint and the starting point divided by the time (Zaitsev & Polyanin, 2002).

There are multiple advanced applications of differential equations whose complexity far exceeds that of the above example in terms of the order of the equation and the characteristics of the dynamic systems. These characteristics include linear, nonlinear, continuous, discrete, and stochastic characteristics. The applications of differential equations have used in understanding electromagnetic systems, Hamiltonian physics, and geometric modeling in computing systems (O’rourke, 1998). Representation of geometric shapes and their analysis in computer graphics has a heavy reliance on the solutions of differential equations with symmetry and preservation of the dynamic characteristics and the structure of the equations such as Hamiltonian structures.

Numerical solutions to differential equations calculated by applying a given set of numerical methods to these equations to find iterative approximations to the correct solution. By definition, numerical methods have an error and may vary slightly from an accurate solution to a given differential equation. One such method is the Euler’s method for solving the differential equation. Euler’s method assumes an existing initial condition and proceeds to find a precise number integer as a solution to the equation. The basic idea upon which Euler’s method develops is a geometric representation of these integer solutions as tangents along a curve that is a geometric solution. The error in Euler’s method arises as the different solutions or curves obtained vary from initial conditions to any given point along with the solution with different variables. The error analysis of Euler’s methods offers several solutions to aide in achieving more accurate geometric solutions (Atkinson and Stewart, 2011).

Taylor’s methods encompass Euler’s methods which are far more straightforward and less accurate than other Taylor methods. Taylor’s methods for approximating solutions use Taylor’s polynomial of a decided degree to develop more accurate solutions. The outcomes of these quadratic Taylor polynomials are single points that lie closer to points on the geometric representations of unknown functions. Solution curves from computational models and those representing calculated solutions using quadratic Taylor polynomial prove the points at successive step intervals in the derivatives are of higher accuracy. This method is more comfortable to apply to higher degree equations with more ease by merely increasing the degree of the Taylor polynomial used in the solution. Runge-Kutta methods of calculating numerical solutions to differential methods utilize a midpoint Euler method with a distinction in how the step size is evaluated. The result is a solution with less error than Taylor’s methods without requiring the added computation of a symbolic differential to the differential equations. Runge-Kutta methods also offer decidedly more accurate results and gained popularity in solutions of equations up to the 4^th order where they offer the most accuracy for the least number of evaluations per step size. In subsequent orders, however, there is a need for more evaluations per step to reach more accurate results.

Numerical solutions used to solve more complex systems involve the discretization of a system to accurately preserve the topological and geometrical features of the system (Hairer, Lubich, and Wanner, 2006). An example of such a system is the Hamiltonian dynamics whose structure can be well preserved in modeling using the Stokes Dirac structure to capture all the dynamic properties accurately and preserve the geometric properties (Seslija, van der Schaft, and Scherpen, 2012). Some specific applications of discretization of Hamiltonian structure can be found in the modeling of control systems where it is necessary to discretize time to preserve the geometry of systems (Haber and Ascher, 2001).

The discretization of systems can apply to the solution of three-dimensional electromagnetic systems. There are significant challenges that may arise in the application of traditional iterative solutions of Taylor’s methods to the discretized Maxwell equations (Bello and Guo, 2019). Despite this, finite volume discretization before the solution of these systems and elimination of emergent linear, large and sparse systems as well as current creates a diagonally dominant and preserves geometry of the dynamic system. There are decidedly complex solutions to these systems with none offering the exact solution without error.

Modeling systems using numerical solutions that preserve dynamic characteristics offers a viable solution to differential equations applied in all domains. Therefore, an exploration of the different structure-preserving techniques is necessary to develop alternative approaches that fasten the iterative process of approximating solutions while simultaneously reducing errors in solutions. This will have a significant impact on computational geometry, modeling 3D systems, and graphical systems. There is a large number of equations that can be assessed to achieve this.

The first set of such equations we can explore for a better understanding in as far as the effect of the use of differential operators is useful in preserving various mathematical properties are the equations of plasma physics. These equations are a wide gambit of equations that each describe a different aspect of how plasma behaves. These equations each describe the structure-preserving techniques of various principles using known mathematical techniques. A major point of reference will be the principles of discrete exterior calculus. By design, discrete exterior calculus offers a reliable method for the preservation of a number of geometric structures. This set of tools can and have been used to accurately express the differential forms of Maxwell’s equations. In equally reliable fashion, the use of differential exterior calculus is also used to prove the continuity of the discretized Maxwell’s equations (Stern, Tong, Desbrun & Marsden, 2015).

Hamilton-Pontryagin’s principle for Maxwell’s equations provides a good way to clearly identify the geometric structure preservation as well as highlight gauge symmetry in some cases. When solving initial value problems, the use of Maxwell’s equations is not necessary in order to evolve the system in time. This can be achieved with the application of curl equations to the magnetic flux B and magnetic flux density D. The divergence equations for B and D are converted into the constraints of the initial conditions whereas the curl then describes the time evolution of the system. If divergence constraints are satisfied in the initial time then they are always satisfied at all times; divergence is constant (Stern, Tong, Desbrun & Marsden, 2015).

Vlasov equation developed by Anatoly Vlasov in 1938 is the next example. The Vlasov equation is a differential equation that describes the time evolution of plasma distribution functions where the charged particles have long-range interactions. The equation first described collisionless Boltzmann constant using the differential of time, momentum, and position. Later, the partial differential equation forms were adapted for use in plasma systems to show that time can be kept constant. The adaptation led to the development of the Vlasov Maxwell equations which when solved using a geometric discretization of the fields in combination with particle distribution functions yield systems with continuous-time and Hamiltonian structure. Original differential systems have some symmetries and invariants. Using conventional solutions, such as the application of Runge Kutta methods on the solutions for tokamak plasma does not result in energy conservation. Using Hamiltonian structures in analysis of these systems yields positive structure preservation and conservation of properties of the Poisson bracket and that of the Hamiltonian value of H (Sonnendrucker & Possanner, 2019). In practice, however, it can be shown that systems cannot reliably have an exact solution for the conservation of both the Poisson structure inflow and the Hamiltonian H but approximations conserving either is possible (Sonnendrucker & Possanner, 2019). The application of finite element exterior calculus in the formation of discretized systems that possess Hamiltonian structure is useful in taking advantage of the existing intrinsic geometry of Maxwell’s equations in all applications. This approach results in differential operators like curl, div, and grad where the properties of the curl and grad are maintained.

Mimetic methods are used in preserving the essential physical and mathematical geometric properties of structures when applied in discrete settings. The integration of any differential is an essential metric free operation. However, the differential form is a multiple vector operation. Furthermore, the differential operators; the grad, curl, and div are all metric dependent operators. The external derivative on the other hand, in mimetic methods, is metric free. Using these methods however, we can effectively demonstrate the conservation of force along curved surfaces and the conservation of energy is certain volume forms (Palha, Rebelo, Hiemstra, Kreeft & Gerritsma, 2014).

Additionally, the action of a Hodge operator, when performed using topological dual grid systems and the operations of a Hodge Star in discrete derivatives is key in preserving the symmetry of the grids. In both these systems, the discrete derivative operator is a formal proposition that joins the geometric boundary operator and is explicit for incident matrices of the selected grid orientation. The conversion rate for Poisson problems in applications with volume forms for h and p refinement results in the creation of indistinguishable curves on orthogonal and curved meshes. The only difference between the application of the primal-dual grid and single grid mixed formulations is in how the differential operator is discretized. However, in either instance, the conservation of the energy within the system and the equations is satisfied (Palha, Rebelo, Hiemstra, Kreeft & Gerritsma, 2014).

An alternative approach that is applied using differential operators when performing calculations involving the equations of plasma physics is the elimination of the Euler Lagrange equations to obtain solutions that are applicable in both continuous-time systems and discrete systems and preserve geometrical symmetry. In the analysis of the symmetry of Maxwell’s equations under certain gauge transformations, equations can be reduced by completely eliminating the time component, for any selected time coordinates. The result of this operation is the fixing of the value of the electric scalar to zero. This results in an incomplete gauge, which means that the result has some degree of gauge symmetry left in the structure. This conserved term is later used to justify the elimination of the Euler Lagrange equations; the Gauss constraint is preserved (Stern, Tong, Desbrun & Marsden, 2015).

It is worth noting here that the variational structure and the symmetry of Maxwell’s equations may be affected by boundary conditions that have been selected. Any boundary conditions specified independently of the initial values used in the calculation in exterior calculus are therefore an important consideration. For example, dissipative boundaries will not adhere in totality to the conservation in energy and the changes in energy within enclosed systems in calculations can thus be attributed to the permittivity of the boundary and not the preservation principle with the use of differential operators. The flux in spatial boundaries is however only possible for systems that have a more complicated boundary condition specification. The explanation for a scenario like this is that the momentum map has a different time specification with such a boundary. However, the differences can be neglected for arbitrary boundary conditions. It is therefore important to apply compatible spatial discretization when working with differential forms in order to achieve the conservation of dynamic system properties (Stern, Tong, Desbrun & Marsden, 2015).

The equations of energy balance for equilibrium in plasma apply a div operator in derivation as well as in the conversion of differential forms in general. The application of the div operator allows the representation of the equation as a general differential equation using specific heat capacity as well as enthalpy. This equation in low temperature and two temperature states can be taken to demonstrate the mass conservation principle in the continuity equation for chemical elements and the equation for ionization equilibrium. The application allows a clear demonstration in the calculation, how the chosen boundary equations as described previously have an effect on the conservation within a system. In the case of the equation of energy balance and ionization equilibrium, there is an established link in the preserved thermodynamic properties and the transfer coefficients of the equations (Nguyen-Kuok, 2017).

The grad, curl and div operator are useful in the equations of plasma physics in that they all feature certain aspects of variation and offer a degree of preservation of certain system characteristics when transforming an equation through different states. Effectively, in the calculation of select dynamic plasma properties, the application of differential operators offers a solution to eliminate the effect of complex variables within differential systems allowing easier comprehension. However, these theories and applications all offer different Hamiltonian forms and adhere to different variations. This opens up room for further investigation and research in the use of these operators in various structures to assess the usefulness and any possible improvements in the application.

References

Atkinson, K., Han, W., and Stewart, D.E. 2011. Numerical solution of ordinary differential equations. Hoboken. John Wiley & Sons.

Bello, M., Liu, J., and Guo, R. 2019. Three-Dimensional Wide-Band Electromagnetic Forward Modelling Using Potential Technique. Applied Sciences. 9(7), p.1328.

Haber, E. and Ascher, U.M. 2001. Fast finite volume simulation of 3D electromagnetic problems with highly discontinuous coefficients. SIAM Journal on Scientific Computing. 22(6), pp.1943–1961.

Hairer, E., Lubich, C. and Wanner, G. 2006. Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. Berlin. Springer Science & Business Media.

Nguyen-Kuok, S. 2017. Theory of low-temperature plasma physics. New York. Springer.

O’rourke, J. 1998. Computational geometry in C. Cambridge. Cambridge university press.

Palha, A., Rebelo, P.P., Hiemstra, R., Kreeft, J. and Gerritsma, M. 2014. Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms. Journal of Computational Physics. 257, pp.1394–1422. Elsevier.

Seslija, M., van der Schaft, A. and Scherpen, J.M., 2012. Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems. Journal of Geometry and Physics, 62(6), pp.1509-1531.

Sonnendrücker, E. and Possanner, S. 2019. Geometric methods for the physics of magnetised plasmas In: Vorlesung/Übung (WS 2019/2020).

Stern, A., Tong, Y., Desbrun, M. and Marsden, J.E. 2015. Geometric computational electrodynamics with variational integrators and discrete differential forms In: Geometry, mechanics, and dynamics. Springer, pp.437–475.

Zaitsev, V.F. and Polyanin, A.D. 2002. Handbook of exact solutions for ordinary differential equations. Boca Raton. CRC press.

Palha, A., Rebelo, P.P., Hiemstra, R., Kreeft, J. and Gerritsma, M. 2014. Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms. Journal of Computational Physics. 257, pp.1394–1422.