In solving a differential equation with a finite-difference method, one computes derivatives with various combinations of the function's values at chosen grid points. For example, the forward difference formula for the first derivative is
f' = (f_{j+1} - f_j})/h
where j is the grid index and h is the spacing between points. The systematic approach for deriving such formulas is to use Taylor series. In the example above, one can write
f_{j+1} = f_j + hf' + h^2 f''/2 + h^3 f'''/6 + ...
Then solving for f' and neglecting terms of order h^2 and higher gives
f' = (f_{j+1} - f_j)/h - hf''/2
Because the exponent on h in the last term is 1, the method is called a first order method.
Write a function that takes the order n of the derivative and a vector terms indicating the terms to use (based on the number of grid cells away from the point in question) and produces a vector of coefficients, the order of the error term, and the numerical coefficient of the error term. In the above example, n = 1 and terms = [1 0], and
coeffs = [1 -1]
errOrder = 1
errCoeff = -0.5;
Solve

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Tim ChrisR

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