As the Mars lander descends towards the surface of the Red Planet, when can its parachute be deployed safely? Open it too early while the lander is hitting the atmosphere, and it could explode – but open it too late and the lander may not slow down enough to prevent a catastrophic crash landing.

There seem to be endless possibilities in this complex puzzle.

One way to solve this puzzle is to use a computer to simulate the landing of Mars, just as the students in 16,0002/18.00002 (Introduction to Computational Science and Engineering) answered this question, their first problem. Was part of the set.

“It was interesting because there are a few ways to model the problem,” says first-year student Andres Arroyo, who took the course during the fall term. “You can model it in terms of how the speed of the lander changes over time or how the speed changes as the position changes. There are different approaches you can try, depending on what your goal is from the simulation. I thought it was one of the most interesting things we’ve done.”

The course, launched last fall, is designed to teach students how computation collides with the physical world. It was developed through MIT Schwarzman College of Computing’s Common Ground for Computing Education, a multi-departmental initiative that aims to blend computing and the teaching of other disciplines.

The half-semester curriculum places programming in the context of computational science and engineering, a field that focuses on innovative applications of computation.

Students learn to use computer programs for simulation, optimization and uncertainty quantification. These fundamentals are coupled with tangible examples that are designed to relate to students who are not necessarily interested in computer science majors. During this fall most students were studying either aeronautics and space science or mathematics.

**modeling real life problems**

“The simulation guys like our Martian lander simulation actually use computers. Did NASA solve our little differential equation? No, I’m sure their model has many more bells and whistles. But conceptually, it’s the same That’s what people actually do,” says Joseph Marzouk, professor of aeronautics and space science and co-instructor for this term’s course. “I work that way, even with my own research. In too. There’s modeling, there’s code, there’s outputs to code, and you iterate between these things. ,

Building the curriculum around concrete examples like this gives students an idea of how many problems can be tackled using computational models. Most students take courses in their first or second year, and many have yet to choose a major, so it’s especially valuable to give them a taste of how computation is applied in many fields, he says.

In developing the curriculum, the faculty wanted to cover fundamental aspects of computational science and engineering in a way that brought the concepts to life for students, said co-instructor Laurent Demnet, Professor of Applied Mathematics, who designed the curriculum with David Darmofle. , says Jerome C. of Aeronautics and Astronautics. Hunskar Prof.

Lectures cover the fundamental equations at work in a certain problem, such as the Mars lander for example Newton’s law of motion, and then students learn to express those basic equations in an algorithm.

“It’s a combination of math with science and computer science. It sings when you put it all together,” Demnett says. “For students, this is really a skill-based class. We want to provide students with skills that can be used later in their studies almost everywhere, and then in many other domains as well.”

**From Equations to Simulations**

During a lecture, Damnet described Newton’s law of cooling (the rate at which an object cools is proportional to the temperature difference between the object and its surroundings). They then ran a simulation using Python code that showed how long it would take for a cup of coffee to cool from 85 to 50 degrees.

He says one of the biggest challenges of developing the curriculum is introducing these mathematical concepts, while giving students enough context that they make sense for some contemporary applications – but overwhelm them with too many details. without doing

In addition to providing solid skills, the examples are also designed to inspire students. For example, a lecture focused on climate science used mathematical equations for heat transfer to refute the false claim that water vapor is a more potent greenhouse gas than carbon dioxide.

But Demnet told students not to take his word for it—he performed a computer simulation showing how greenhouse gases affected the overall rise in global temperatures over several decades.

Outside of the classroom, students explore everything from optimizing the placement of cell phone towers around MIT, to evaluating the impact of geothermal heating, to evaluating how COVID-19 vaccine effectiveness decreases over time. applied his computational chops to a wide range of real-world problem sets. The system can be at an indoor temperature.

For Penélope Herrero-Marx, the geothermal example piqued her interest as she wanted to someday install a system in her home to reduce her carbon footprint. Herrero-Marx, a sophomore majoring in mechanical engineering who took the course last spring, was drawn to its relevant problem sets, even though he had little background in using computational approaches.

“Some of the problems were a little scary at first because they were so big. For our first P-set in orbit we should model the landing of Mars. But the professors did a good job breaking it down into smaller problems. Don’t get overwhelmed Every major problem can be broken down into smaller problems that you are actually capable of dealing with,” she says.

She is now sharing that knowledge as a teaching assistant for the course.

Fellow teaching assistant Mark Chiriac, a sophomore, took the course in its first iteration. The math major wanted to learn more about algorithms, but also wanted to focus on interesting applications such as planetary motion.

While one of the most difficult problems involved locating cell phone towers around MIT, it was also among Chiriac’s favorites because the example was so realistic. Successfully solving that optimization problem gave him the confidence to apply those skills to other courses, he says.

“This course puts parts of coding, math and physics together in this beautiful mix to give everyone the tools to tackle the very relevant problems that are urgently needed in our world right now. It showed me how to use science.” These different realms intertwine with each other, which I knew existed but had not yet experienced for myself,” he says.

Ultimately, the skills students build in this course will help them tackle the problems of scientific prediction, whichever way they choose, Demnet says.

“I hope students will walk away with an appreciation of how computation can be used to simulate really complex things in the world around them,” Marzouk says. “I hope they see the power that he has and have some appreciation that this isn’t just a black box. There are really interesting ideas and algorithms that explain how this happens. Whether they’re going to spend the rest of their career Spend it learning about those ideas and algorithms or they stop here, I think that’s a valuable takeaway.”