Trapezoidal method for NDSolve?

Questions
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1

May I ask if Mathematica provides a trapezoidal method instruction for solving differential equations? I checked the reference materials and for NDSolve, it seems that only explicit Euler, implicit Euler, midpoint method, etc. are provided, but it seems that the trapezoidal method is not provided…

2

For Heun’s order-2 method (a.k.a. “explicit trapeziodal”), that is the order-2 "ExplicitRungeKutta" method (which has an embedded error estimator):

Method -> {"ExplicitRungeKutta", "DifferenceOrder" -> 2}

NDSolve`EmbeddedExplicitRungeKuttaCoefficients[2, MachinePrecision] will show the a, b, c, b_{err} matrix/vectors of the Butcher table if you want to check. The last vector b_{err} is for estimating the error.

Alternatively, you can code your own Butcher tableau and plug it in:

explicitTrapezoidalCoefficients // ClearAll;
explicitTrapezoidalCoefficients[2, wp_, caller___] := 
  N[{{{1}}, {1/2, 1/2}, {1}}, wp];
explicitTrapezoidal = {"ExplicitRungeKutta", 
   "Coefficients" -> explicitTrapezoidalCoefficients, 
   "DifferenceOrder" -> 2};

Method -> explicitTrapezoidal

For the implicit trapezoidal method, you can use

Method -> {"ImplicitRungeKutta", 
      "Coefficients" -> "ImplicitRungeKuttaLobattoIIIACoefficients", 
      "DifferenceOrder" -> 2}

NDSolve`ImplicitRungeKuttaLobattoIIIACoefficients[2, MachinePrecision] will give the a, b, c matrx/vectors of the Butcher table, if you want to check.

Alternatively, the Crank-Nicholson method (a.k.a. “implicit trapezoidal”) is coded up in the tutorial NDSolve Method Plugin Framework—Wolfram Documentation