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The elements of an arithmetico-geometric sequence () are the products of the elements of an arithmetic progression (in blue) with initial value and common difference , = + (), with the corresponding elements of a geometric progression (in green) with initial value and common ratio , =, so that [4]
The first step is to triangulate the polygon (see Figure 1). Then, one applies a proper 3 {\displaystyle 3} -colouring ( Figure 2 ) and observes that there are 4 {\displaystyle 4} red, 4 {\displaystyle 4} blue and 6 {\displaystyle 6} green vertices.
For such problems, to achieve given accuracy, it takes much less computational time to use an implicit method with larger time steps, even taking into account that one needs to solve an equation of the form (1) at each time step. That said, whether one should use an explicit or implicit method depends upon the problem to be solved.
How to Solve It suggests the following steps when solving a mathematical problem: First, you have to understand the problem. [2] After understanding, make a plan. [3] Carry out the plan. [4] Look back on your work. [5] How could it be better?
The Crank–Nicolson stencil for a 1D problem. The Crank–Nicolson method is based on the trapezoidal rule, giving second-order convergence in time.For linear equations, the trapezoidal rule is equivalent to the implicit midpoint method [citation needed] —the simplest example of a Gauss–Legendre implicit Runge–Kutta method—which also has the property of being a geometric integrator.
The solution is the weighted average of six increments, where each increment is the product of the size of the interval, , and an estimated slope specified by function f on the right-hand side of the differential equation.
(Figure 4) Solution of ′ = computed with the Euler method with step size = (blue squares) and = (red circles). The black curve shows the exact solution. The black curve shows the exact solution. The Euler method can also be numerically unstable , especially for stiff equations , meaning that the numerical solution grows very large for ...
The application of MacCormack method to the above equation proceeds in two steps; a predictor step which is followed by a corrector step. Predictor step: In the predictor step, a "provisional" value of u {\displaystyle u} at time level n + 1 {\displaystyle n+1} (denoted by u i p {\displaystyle u_{i}^{p}} ) is estimated as follows