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= What is the difference between an implicit solution and an explicit solution? =
The implicit and explicit solutions are numerical methods allowing to move from a continuous to discrete model of the world in which quantities are computed at a finite number of points in distance and time. (Meshmesh or Gridgrid). Therefore, approximating the partial differential equation by a system of algebraic equations.<br>
The mathematical description of many physical processes involve differential equations. In the case of the shallow water equations, the rate of change of depth and velocity at a particular point in space and time are described as functions of the depth and velocity, and the spatial gradients of the depth and velocity, at that location. The solution evolves according to its current state. The depth and velocity at some small time increment into the future can be estimated from the current values and their computed rates of change, but here is the problem: the rates of change keep changing and are a function of the solution as it evolves. The small change in the solution that occurs during the time increment influences the derivatives, and therefore the change in the solution depends on itself. The bigger the time increment the stronger the dependence. There are two fundamentally different approaches to finding solutions for problems where the rates of change (in time or space) of a set variables are a complex function of themselves.<br>
The first (Explicit)explicit approach is to estimate the future state based solely on the current state and the current rate of change. The simplest approach is known as the “forward Euler method” which is easy to understand and implement, but often unstable for systems with little or no energy loss mechanism, and even when it is stable the solution is usually quite sensitive to the size of the timestep. Higher- order methods break the time increment down into sub-steps in order to attempt to account for the “the rate of change of the rates of change”. Such methods include the second order Euler and the fourth order Runge-Kutta method (which is what TUFLOW HPC uses). Explicit methods attempt to capture all of physics in the solution that the equations admit, and often will become unstable if the timestep is not sufficiently small enough to capture the physics accurately. Hence explicit schemes in fluid flow problems have to adjust timestep according to flow velocity (Courant number), wave speed (Celerity number), and diffusion speed (Peclet number).<br>
The second (Implicit)implicit approach is to estimate the future state based on the current state and the future rate of change. There are two common approaches to this “circular reference” problem. One is to reformulate the equations as a matrix problem, and the other is to use an iterative approach whereby the future state is repeatedly updated after successive calculations of the rate of change at the future time – also known as a backward implicit scheme. Both approaches can be very stable and enable larger timesteps, but the solution is permitted to “skip over” physics that happens on timescales smaller than the time increment. The timestep must still be appropriate based on courantCourant, celerityCelerity, and Peclet numbers, but due to the iterative nature of the solution the time step can often be 10-20 times larger than that required for an explicit solution.<br>
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